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As you can see here we're talking about
finding the area of a region bounded by two curves. But as we get as we get started I want to
first discuss
or do some introductory work that will
accused murderer that will make our companies here not only for the section
but for sections of the future
so
i was the first no
how would you guess how we find
they hardly
of a particular rectangle
yet so let me just give you a little quick example here
so that uh... what i could say it more generally
suppose we have of course it looks like or taxes
it supposes the school it is for the support to businesses for three years
here this is the delay is that this is where they get to it
in those will lose
we want to know the height of the very typical little girl you here
done in this case of course we can count but i a greater down here
uh... worthy retrieval begins
there's one year to year streets
good note here
when we do this
it will take the largest why value that's right here
answers cracker the smallest y value ac
and of course i throw here they had to
it was that once the credit union to hurt positive three
chloe seed capital inc this friday
miscellany too
we take orders
acutely
orchestra wa right
so prior
smallest y
solely general
okay just in general
if we have some kind of a rectangle sitting on the x_ y_ axis
kahi
viewing recognize that this is silly
uh... they top of the rectangle if you will
and that's a really wanted a license to
and the bottom of the rectangle as it was super woman
or execute g
uh...
largest why are you then this product can be calculated
in this fashion simply
why i said to marantz writes a book
will receive a need to be in the future also
peek at the lake the high for the lady rectangle
this was on a local of word for it though
so we could go through the same kind of argument
it would be difficult for us to make this decision
but suppose we have a rectangle
there is going on inside
denim
for intellect this
so this link
cabling that we're talking about here albany we calculate this link
creating literally for
would be the largest
x value
will be the largest of its value
local access to
here is the smallest x valued acts of war
so this
as we're reading from left to right we're going to mexico on the execution
of that link will be the largest expel you subtract
the smallest expanding
and that's how we would have to make political activist
well that this is going to pay off worse in the future
uh... indian affect what i want to be next to them not to review how we
calculate the area
uh... under occur
and remind you the idea behind it and what kind of
smooth and out and radio freaking forever will be paying for the future
ok solo contracts to find the area of the region bounded by white will export
first one
why does the role as a line which is horizontal line
and uh... exit polls the rolodex equal to electrical zero mexico to has line
sort of vertical you're talking about and an area a region that's influenced
by the particular pieces
actually remember
in the past
collegiate charter a problem with this is an issue station grab for well region
we're trying to find the area
so let's do that will quickly union discussed further
here so
was supposed to
along two three four five the just about perfect
willman clean
three chloro climb
and uh...
i want to do this for a bit early acceptance so that we have the same
scales there's going to be one
and they're going to be treated as just so that it will be greater sketch
well we don't know why blacks for was formulate parabola
that uh... uh...
in insists protection and one unit
here so that workers would be here
the end and we know that witnesses the all along that what i would be once were
plus one which would be too
he wanted it to the into will be to use for your call upon what you saw
it's only in that
it with the parabola localized first
on the right side of the except by the white axis parkway
now remember
boundaries
okay so this is the corona calypso platform lucy
there's also been alluded to
myelin equal zero which is nothing more than the x-axis
inoperable and icicles zero which is here
and of course it's equal treatment is here
so we're being asked to find that the stereo pan decline of those
four life sir kroger's
and it's not the moral of the area and the recorder or
so you'll recall when we do this
is that uh...
we calculate the area using your
it would be in a girl from the los textile usually rode laxatives are used
to
of the function himself x where roselawn
with respect to be acts
now when we first learned how to do this for we really
worrying that uh...
introduced to with the idea of breaking this region into multiple rectangles
yay
foshan lots of rectangles
in what she had done this year
available development messing with the work that out uh...
any
building i'm always rectangles
and uh... we we uh... summed up the really shows rectangles and then we left
but rectangles did
in smaller where
so there were more rectangles
and literary deals give even smaller with so there were more rectangles and
of course he signed the rectangle was got smaller and we had more experience
to work from
initially that there are a part of a ship to the area
and so we left the within those rectangles
go to the world by clicking on it
is included be convenient here to recognize the leadership says the thief
republic
does not require executives i was before
soul missus what we're talking it over to the region that we're talking about
was slipping one single representative rectory or
annalisa saluted is infinitely smaller what
indian louis control here
in end in recognize of course
behind at this rectangle
is nothing more
dandelions defined by this forever
and that why have courses elsewhere plus more
sold that why value is the height of the rectangle
and deliver the rectangle
we called the it company
well with the district will be exits
isn't x belief
but it's infinitely smaller is going to be infinitely small
insult me it would come up here
and think about this too liberal
list for of the individual
is the high of the rectangle
in this part of the integration that you're going to set up
is nothing more than the water
influent and think about that as a product
of the high endless
so that the other crazy area of that one simple example
no
and uh... this is not difficult for us to do it
but uh...
i want you to think about
powell were wonderfully not only for calculating these areas
but this will service
uh... great deal for later for the use of this contest
course polluted
because right now for the second let me get custody
to uh... what my throat
initially i would like that so i'm going to remind us as we do these things that
many times
uh... you generally need to think of definite integration
as a way too
infinitely saw maria's
and infinite number of things
in each of those stains
kessen infinitely small measurement
truly case here's the area under the current
were any were were we haven't included a number of the rectangles
inside a regent ward connerly calculate the area
hit me
we represent one of those wrecked ankles
all that rectangle has a measurement of with that is infinitely small
because that's what we're going to happen is we're going over a period of
six months or what
in as the west with interventional the number of rectangles it's infinitely
large events howie's immigration to enter
their infinite number of rectangles each which has an infinitely small white
now
that we're going to use equation to do a lot of things
if we can think in these terms it will help us
bill the urals to solve the problems that we want to sell
but this section is about
finding
uh... areas that weekend kare
which is so
will begin on the next
page
with uh... an example of that using this concept that we just talked about
okay you can see the problem that were of
networking on here
is stated in this fashion fine the area bounded by the graph supply pls forgive
ex-wife wexler
now e that many times the hardest part of these problems is that true today
what is your spoke today with a little planetarium
but what he really is that we used to have to prove given to us so
i'm going to give you an idea the way we look back at home and act like im really
no warrant building its own thinking okay will
when you look at these two graphs injustices rough credit which in
addition
that uh...
when
this is not really
uh... the way that i would be finish this houseboat what you see my figure of
the way you may think
well let's see what it calls for this
is a critical of something like this
in my intellectual areas that were at the look something like this
pinellas through what
they were supposed to find the area bounded by the few graphs
so we have to recognize
is that we're looking for
aim region that is in close that is surrounded by these graphics
if you notice here this is that region that is surrounded by those preps
and so
that's the area that were being extra fine
now once we understand that
and we need to be a little more careful by the way we put together a graph so
notice that i don't really care about things that are in the second quarter
this entire region are fighting area falls
within the first floor
in
i'm going to be a little more precise now as i recall the graph
day of the region that we're looking for
as usual grand and thinking that figure out where this intersection point years
of the nation
well the israelis are accessible
and uh... that i know that
of listless just think about this
people we could do to find out where those fiend
grass intersect
in the way we will find out where the two graphs intersect
is finding out where
exporter
if people do this for a tax
here so if we saw that equation then we could do it
uh... now
there's uh... that this is a quite as easy as it may sound whitman i really
need to go through this trouble
well let's suppose that this is what we're trying to do
well we would like to be able to hear it up
of the sort of extra effect within this looks like this is the actual requires
some sort of ecstasy pills is the world
it would be settled at six where ninety six for the white house power is equal
to zero
and all that will allow us i suppose
uh... to after this to some extent
but i don't know how much did nothing to do this with said okay the smallest car
bexley contractor has excellent have power
incidentally behind extern xd lol
three half-hour
because normal why actual one-half of exit freeh as i get back to the floor
have supported
minus for peoples the world
okay right now
at our how does that help us well what causes that to be true
well excellent have to be zero
which means the square root of ecstasy or or which would mean excessively zero
and so that's the steps value that we're talking about there
was the expanding that we're talking about here
all their own happened when xa three have power
minus one is the zero significant package
and we would send each of those factors to the world
well
how can the exit extra power be equal to more or less let's look at this little
differently what possessed of the free hasta army
well that's really
elsewhere reading of exchanges in it
for the spirit of extremist wants the world
or the story of exchange is equal to the war
in west europe and the orthopaedics infotech mister roda factors were senate
district would have been proportionally more extreme disturbing was the facts as
the four
and so we find out that this tax value has walked the excited here of course
where he was a hero
so that allows me
to make this breath a little bit better and i mean i suppose that i could have
done this school differently anyway
so this is where we want is a wide access
when breathtaking liable explorer for instance
z one xa zero why would be zero in on taxes one why would people that that
would be these points
in seoul
background for looks something like this witness
and
and it's going to go on from there and cool this direction
i'm black and white illustrative x his exit zero y zero
on the other hand
if x's and on
why would be the square root of all of which is warren so
it also status for but that graph
of course is for a graph looks like this doesn't and they don't go on from here
so we may have been that without all the work that out the here
it's because they're arledge of racks
now that's the reason they were trying to find
uh... the area
here's the approach that i would like you to practice
roaring representative rectangle
from atlanta representative rectangle
we can calculate the air
so
bustles through three thousand one representative rectangle
inflicted extending it here
with espy infinitely small island nation and would only be west image
and so we're talking about the wisdom mentioned being
an accident shin
then as infinitely small i'm going to call that d_a_x_
the height of that rectangle
is the next question
in america where we find a hike the rectangle is we take the largest
wytheville
all in this case the largest y value depends on the curb
does this worry directs
so the lawyers one of the aren't you
is the sort of acts of crap
the smallest what about you
well it's not for about eighteen happens to be on the car
why people x lawyer
here so that small swap valued at six sweater
so that is that many of the heart of the rectangle
so the area lawrence fall rep wall
of the
small
rectangle
okay
would be
gate height which is the story of the tax minus six later
times would which is being x
so that's the area of the tiny rectangle
we haven't yet for that number of other rectangles
and so we're going to the uh...
all of the areas of each other examples
he gave the people there
but as we've got an infinite number of the home that's the way we use
integration
now the first rectangle week so i have will start at the value exit polls we
were
so the very first article is going to be starting right over here is
and the very last rectangle we're going to have orthopaedics is equal
insular adding up all the rectangles between exit polls the rolodex equality
is not a rectangle superiors
into their share his if this were to classics slaughter
detects
and so from the rear then we can calculate the area and i'm not going to
go through that
uh... the trouble here but i i guess i will in this example but in the future i
might have
spread this around and just give you an answer
and i'm going to write a story of expense tax for the power one-half
so that they usually anchor
they had to do a definite integral
uh... we first figure out the indefinite integrals leah definitely will address
to the white house power when we did we increasing its fun about it on
and we will find out the recess because unit for
so
uh... will only increase one-half by one we get for you have to have post-war
history has
in we offset that by the recent before the actual which is two-thirds
and of course we increase the actual arrive so we're level and we get extreme
and we offset that next on it with a factor of one thirty reciprocal
infront
clue as to the indefinite integral
now that's what we have to reevaluate for the next phase he wrote it so he
walked
but remember we for sure student wanted for the axis
and then we substring zero and three eighths assoc
overstuffed rewarding and will have two thirds
times went on to the jury has power
minus one third
times go into the third power
immune from at least have cracked or we get on the substitute zero and corrects
one twenty to thirty five zero to the three harrahs
minus one third avenue ste
zero to the power three
acted out of course overhear zero in eight hours ago
and so all of this far interviews being single
and so are real issues but we have here
well warren beatty power is workin
so we end up with a two thirds times one of the first position
minus one third time for the second position and of course two-thirds minus
twelve thirty seven four one third
here we're talking about
uh... area
so one-third of sporter review whatever units that measure
so
that the most important part was not the integration arias how we set out the end
of how we built the integral
by thinking of infinitely small pieces
really
an infinite number of that we had uh...
by using integration
said we'll look at india second example here in a moment
let me uh... let me close temporarily
as we go to that second problem
intermixed example
unsinkable being a student
on the area of the region bounded by the graphs upwind collapse or close to where
it's
what the effects of store
the vertical line exit polling before or after six-foot-two
kahe once again p
setting up the problem is not usually too difficult
if we know what it is were actually trying to accomplish in moderate or
talking about
so we thought we'd always have to spend some time looking at a rack
uh... of the region
actually hills trading the region
soul
but let's do a little bit of work we recognize that white votes were close to
a hex
incredible
in here
a couple of things that would be nice to know that the struggle is worth of her
critics
interacts intercepts
select their sister continue to decline
missus intercepts
are uh... big x value the cars is quite a bit easier
so would be looking here
assuming y zero y axis because that happens we'll see
console equation very quickly
because of the right side factors actually
yes we would get a sequel zero net siebel negative too
as bs intercepts corporate law
how else would we need to know how the projects
another therapist
you know we have another struggle separately
because it has a positively coefficients uh... in front of export
uh... the vertex is a
local maximum oral monotonous visible now
and so
it would be useful to find
lifetime
uh... so that we did find out where while privacy world actually have a
and uh... uh... projects local memo accumulative mineralogy match
so what time is a lot to star-studded
is aware that zero
well math zero minutes of the political up
so over objects
okay
has an exciting to you on and we would simply cannot calculate the wife value
micronesia equality into the function of course
and we would end up getting caught why is equal to
one minus two or maybe it won't
so the perfect apathy a day he bought
now
well we're here on thinking
well it looks set up
to uh... princess
also seeing
heretofore it won't be reported on
too
through the
actually i think uh... i don't want you to reassess status
uh... was was make these indian institute is focused on
floor
six
and i think you'll see what i've done this year moment
and uh...
of course was snake
this negative too
in this eighty four
and uh... of course we needed it goes in eighty four four excitedly awaiting a
positive
of ought to or are
uh... maximum its value
closest to prostitutes
okay so the native-born
anyone will be approximately right here
which is called it
and uh... we decided that
it crosses the word extremely attracted says the rule
and also an eighty two
when exteriors ninety four
well why would we get here will look at what
in eighty four squared plus two times eighty four
which is sixty nine dash eight working it
his that's really the reason that uh... conductor
annexes causes it to
canadian
than what we want to be in a way to calculate that really easily while at
the
uh... elicit i'm sorry federal signal up there
why would be uh... to use for air
post-season student
so that's four plus four or eight
so it's a here
usually gives some rough idea
aquat dot parabola looks like
in the
playtex people are interested in
what does this work
well one of the others for like little x calls for our streets always line
that um... crosses the accident you know why i had to support us
but it is the lowest four
and we know it as a negative slope
uh... than anyone still open so we could calculate some other pieces
very easily but was it was just uh...
let's imagine effects was uh...
dat it for
and uh... events is maybe four
then why is that negative x
plus for which would be a
so little line goes to the will also
malcolm x_ c_n_n_'s
positive to determine why i'm not worried about that because we're going
permanently forceful policy too
so this is part of it too
we would end up with y is positive as well
native peoples four
so it goes to the spot
so that is as a rough sketch
buses or one o'clock close
remember were relieved to know
the
region
the area of the region
talented by pieces of blackouts
hand
blind x people maybe four
which is the story
inline vaccines or positive too
which is a small
and so that really is is this little piece right here
so we're looking at if you will
the area of the region
the action in reliance there
and you see that that really have repeatedly broken into two parts is that
when i call this region
hysteria
we might say is faces lol
enlisted smaller
region from a safe area is a secluded
we can't calculate
the colder air travel and it will ever really appreciated
is going to be the area of the first part of the region
excellent both the areas that the prayer that region executed
to calculate cases norman bay area of the first freakin
we're going to integrate we know that
and we're going to let's let's look at a representative rectangle in this region
representative record or
now we need to know behind phil gramm record
in seoul
uh... in been admitted this representative rectangle
the highest harder
it is and why they are used to find a byline
and so that's negative ex post four
in the lower part
calif rectangle any rectangle in that region
is a warning values defined by the purple so why does export plus two acts
i can remember
the height of that rectangle dan
is the highest
minus the lowest
and so we would be in a grade that the higher tax
uh... that rectangles i truly understand that
so i don't have a rectangle
is going to be
namely the x files four sa hi strongly
minus the lowest walkway x where it was two x
so that's the heart of the record
nammal with the district attorney will remember or making them infinitely meryl
in the ex-juror action
basil weapons and really roll
and that's why we collect the acts
and so that's the with other infinitely
represented represented rectangle
barely rectangles we're going to start all over exit negative four
but she noticed
that mainstream after we go past let's pick a color position right in here
instal
when in the middle
well x value that is
because when we get on the other side that except for you
knows that the parabola is on top in milan in islamabad practical is on top
of my pocketbook
in so we have a change
in what the highest y
and most why it's a little bit of a difference attraction
uh...
and we need to find that point of intersection
k
before after
animal it after we do that will have our limited integration set
well work in the grass intersect is where there are quite axes cause the
right to be the same
in seoul
uh... want has a lot of extra close to that extent will know
weirdly
and is the same as lying on the other program window on the other grabs
neighborhoods forceful
so you know when those two reporters solutions to the equation
warily seriously and exit to prep for on both sides of exporter
history it's not a storybook weaver was led by the way wage
in this case i think we can factor
the left side
uh... how about x
before
anesthetics
uh... minus one let them do it
initially get siegel nineteen eighty four
connects people positive one
negative four is the yes i do have a right here
negative she made a positive one
is the excel you that we have here which is the other luther section so those are
the people at that intersection
so this upper limit of immigration isn't this valuable
well now we ought to be able to do this in a growth fairly routinely
okay
insider
we actually had a native explosive for
madison exporter
in montana state relaxed we distribute their out
negative side up
and of course
ends up being uh... the fact that we're going to be in a great walks
uh... creative exporter
minus three x above four
from negative or positive toward
and uh...
placebo effect relation to get here
only giving indefinitely to rule petroleum exporter there will be one
eighty one thirty excuse me
inland minus
treaty
has exporter
employs four x
and that we have to evaluate from eighty-four of equality
was being a little more work than i was anticipating
superfluous if a candidate may give you that i actually had warned for our
accents and we didn't get it made me want a fair
minus three has
thought four
and from that will subtract what we get with substituted in the eighties
for all of our xd
in that case
well making one-third comes in eighty four she
which will be here positive sixty-four or for any willing
and you won't have my eighty s
three have signs of negative forceware
when needed
training before we get sixteen
and so uh... the uh... what freehand from sixteen
for making oracle
and then finally we have a
plus four times in eighty four alerts in eighty four
but for the world i'm sixteen
work tel aviv calculation silently of traditional
okay
console my calculation tells me that that's a hundred and twenty five or six
remember less area of the first reach
now uh...
but for the next stage
and calculate the area pacific region so we can make the total area
and uh... let me kind of three sixty four f over their quickly
but there have been a real we'll schedule it for the overseas last date
*** here
uh... in in it's not the scandal but i wanted to make an afternoon black
actually get this party here now
before we go on calculating the area of the region
uh... in close innocent parts of their you're calling a such too
i love point out to you that in the first part of a problem
we in calculated
media i have this rectangle and we didn't really care
where the retrieval the or areas
here we have for the rectangle above the excesses of purple the exceptions
all the way we care about
is taking more important value of scrapping the smallest what about you
to actually illegally electrical
and that doesn't matter of whether the rectangle
is although they accept as all the essentials
or some other symbol
which individuals is a little effect occurred about winners regional markets
not like what we're doing we had uh... the area under occur
and we worry about sometimes in heated valley for our area
we're just finding a positive height of that rectangle
and uh... said that as an important concept a relapse
okay will in this region and we look at any representative rectangle
fairy
so
well we need to do anything
i remember
this is where x is the will go on
and this is where x is equal to goes level we have on the other side of the
other page
uh...
so we were going to set up amy
interval here they calculate the area of the second region
in this case though
for women in our growing roster rectangle of the neck so i don't want
him
and with server rectangles and excited
but also what changes the reason we're doing this in two parts
is the hardest part of that rectangle is defined by
whiny all
elizabeth rather like a last word list index
so we were fighting the largest y
is x where close to acts
informed that we subtract the smallest hua hai
anastomosis why is the largest defined by that straight line
which is needed expose will
and forced helix
so once again this is the highest
n
and this is the layout
of that rectangle normal bundled together we get the area
and so we have
the area of white representative rectangle
billions infinitely narrow
inwardly interval is more an email all the infinite number rectangles that we
can have but we wanted to
and um... the works similarly to what we didn't make it will be the first thing
without paying
him in this e-mail will or actually agreeing it was simple client-side
with events where
plus three x minus four are
and uh... then of course
bos
program
when we finally indefinite integral which would be one third execute
plus three have six where'd
maheshwar x
and we evaluate from one of the two
laurin with four here and i'm convinced that you could do that
than we do this calculation
we end up with seventeen over six
so remember that total area
his senior year of each of these for relief
and so the total area
would be even supporters accepted
in this case for seven hundred twenty five or six
plus seventeen or six
within the the of
one bullet to six
an apple gene balloon for each unit is six oh seven and one or three d
uh... square
units
found act after using if you remember you saying how these things work we
could have done this caller id mon-sat
uh... if we were still had to intervals
but uh... it might have been a little easier and from the very beginning
uh... intended
anyway out
but was look at another note here will need
separatism and out
case you can see here that the trying to make
and and i referred to her
of the the last fall
that this new method this new idea so i really do but
that and that method being
look at a representative record
so they're better represented here
it
and when we look at a representative rectangle
if we calculate the height of the rectangle like taking the high school
and i used to have people over for about two
business method will insurer
that we don't get a negative result we calculate area
and as you recall up all my *** here
uh... i is needed for about uh...
as most of the things were a heck of a x's any of this works is b
we have to worry about what you get the herald for captivating hysteria
that because the area why well all the x-axis that
and we thought we would get his name
but here
at this for
is defined by f_ of x_
well
what we would end up doing is we would calculate the integral
asserted excited at the indian detects that any
in our representative rectangle
has a high
well the highest lie about you
is always the same
okay
behalf why a value is always on the x-axis
and why values on the x-axis thursday or so the largest library you can talk
about rectangle
as is that why it was in europe
and the bottom of that rectangle fleet at the highest winds cracked the lower
four
all the bottom of that rectangle is the law and weekends
from whatever than definition of function is
so this
it's the heighth
of that record
and of course the width of the rectangle
the ac
w detects this is a bullet
although this method
is going to ensure that we calculate their yeah
but actually it wasn't even be negative
f_ of x_ kleenex
it is the same as the negative of the world for me to be
uh... x bs
in seoul
this negative uh... we see you here
insurers
that one-week activate this central unit unit but we're taking the opposite of
back that's why is that the carpet
our new technique and later we'll look at this
by this actually captivating the height of the representative rectangle
will ensure that our result is always trust it
in legal way where utility
uh... probably looking up although
sitting there and all those firmly above earlier little bit of a list of through
there
that the except to say it is
always capillary hearts we take
heidi fleiss value
will apply value
and uh... that's the way will calculate that higher lai values depend on the
function
the artistic black for acts
folk enrollment vagueness problems but only on the next page uh...
an excellent indicator place because the standard way causes a little bit
theories here
who as you see here
we're defining area bounded by
the but once we recall one of my sense
the growth of two article exposed to
and there's progress is only a few uh...
one but unlike or exposed to ripples strangely
of that more seriously is the fact that we have life story altering
what specs
and uh...
nowhere that scope it would recognize from
pre calculus that exactly a brad
letters although the laughter opening prior rather purple location
occur again
or support law orientation
and uh... equally knows a lot of trouble to figure out of the organization that
lee with me
'em
not
let's let's just stop thinking about it for a moment here
program this
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we could just think about plotting some points
and as we do then we would come up with some kind of a i'd be here
on what this might look like
and so for instance we could choose index value
of let's say it's legal
well so-and-so one
genocidal
well them twice where it would be one minus one or zero
and so that they also want to use it
now one-line to twelve years we can have an experimental
prisons the tax was too
women have watched where it is not much to watch first native-born was must wear
a wire i think it'd be a good value
so
one is the largest valuable
and how we could use a recipe zero
and then twice where they put the wall
and that means why deteriorating plus or minus one
so we actually have a point is the local one m two point zero carbonated water
both under stress
what's up
the latest round of stevens along i i i suppose up if we were actually equal to
uh...
negative for me
then we would have launched warrior is one months leave three which is a four
and so on i would be possible initiative
you know jump clear up the day-to-day disrepair consul wife's where it will be
one minus database
which is not a salah i would be or smart mines
will be equal to
plus or minus three
peso
what's up
was breathless parabola
none of them privately uh... perfect in my brain off
burden of at least at something like this weekend
convinced gardena give some idea what this graph looks like it
hits on one thing work
let's say
this is where we want is for x
and of course in that case placing the elisa
and one x one zero
wanted to be housing loan
bday line could be native-born
so she will come a long time but i happened to your comment he was on a
graph
in a week scene at their will
in the opening to the left is
and i know its ah...
uh... not just any old below suppose this is what he had eight is
him so he wrote the worked late
you know you would be to limited for any he will be roughly causes ring
and uh... sylvania date culminated in three years on the graph
insulates the about it here
middle it that but what to do at that
and course
income appear like this as well
but i i don't know where i had
uh... that was let's look at this line
okay
let's look at uh...
to lines as he goes into exposed to
we could easily solve after why bible printed by one-half providing
of both sides back to you uh... prefer to think about what prompted by one-half
so it wasn't for the one hand explosive line
and then we do you recognize that forces line where the white intercept
is it lol
so those who at that point
ya'll
it might need useful to notice
the clinics is negative any
the unwinding no serious x's native any
underlying it won't have time to make it any
postal
an estimated four was swollen which is to make it for me
so they go through this again but there
you know i was fortunate to live but
so
here
loess
play like that a straight line up college and i think those are very
strict
so we're trying to find
the area of this region
the problem
is that when we started worse
we'd know important pieces here
but the problem that we have
his interest in philly roller represented the handle here
lyndon b part of the rectangle
is defined by the straight line in the bottom of the rectangle is fine but the
parabola
but as soon as we move past six equal zero that is as we get to this part of
the region
work harder and elizabeth i invited for apple
and salutes the bar
and so they calculate the height of at rectangle
isn't there for calculation
then the highs of the rectangle first portion
so i'm going to come back
and think of this
uh...
region having area is a blind
initially in the area basically
and sort through the area that working for one sixteen
is going to be a case of one thousand two
now yet
we have we still have a little bit of work here because this is a strange
crawl major delays to find his little the functions of the year
the grass for a little unusual to ts
so
here's what we have to be concerned about
uh... for a civil war
okay
we're going to set up a memorable
in the inner role is going to start with rectangles
one over here were actually thinking about eight
sore beginning lyn immigration is that it will be ready
but need rectangles only believe bill clinton forex as include zero
because lately country at the higher the rectangle changer creek
yeah
the top why in that region
so the largest while ella largest body of the fine line does not
ended that benoit and
we cannot do a lot of those riders
that y about it you you one-half ex post war
minus
ne small that's why it happened
is right here is defined by the program
and here's the problem
orleans that's smallest why a m
and at that
it and then that we have to think about this little bit
i remember that brown was defined by whites were people one minus tax
and were looking for the smallest why won't you were looking for a wide
mystified by the parabola
well
itself kinda mind here
well disposal watched where salon you remember
is plus or minus supply susan isaacs wanted people to minus the florida one
one six
in an excess of what we did a little earlier
because when he shows index dot here
to realize that positive and negative
so it makes little sense about what happened
now is the problem is that all
wishes it's in this particular case
worldwide piercing the models
honor malignant
qantas
ones up here
any
positive
well which one if you were interested in it were interested in the naval soulful
for of swore that lois one evaluates the native of the so everyone like this it's
not just the squared assistance
so we're subtracting the lowest while valuable as one italian demands for
everyone process
and then the extreme member
is that infinitely small limits
so here is the height
and the rectangle here is the way i have
and so we have a high times with if you will for the area of that one
representative reputable
last month it a little bit of work for us to do
and uh...
right maybe i will cancel or when in fact i wouldn't listen to take some
something
calculations force
both see that
solo
cases warm
his hero from native eight to zero
our
inner ear and that is what we're integrating we need to set the fire
so we have any want have fanatics
across a wall in
subtract a negative spirit one ninety six
so there will be able to tell us where we don't want my sex
proper act
we're going to do a little bit of thinking here
can come up with it so
indefinite integral
canyon that marijuana half actually do
will increase the extra mile low-end
and uh... mall filed by the recent will then do it so i will only increase uh...
to the next where do we have more power back pain might have
and we already have a one half so that's the one fourth
exporter
intake agreed in a row quickly recognize we get one half its which would start
with
mineral water of course is just x
now we have are
worked out for us
and so although i don't have much space here
political richter thus far
and look about how would we calculate
the integral of the squirrel letters x
com eventually i would expect you to be able to do this you know the success
final results
but that alone would normally be here's we think of that as one nine six of the
car what happens
and then we would use the use of the tradition that is a great you can be one
nine six
that means we have the internal revenue to the one-half the limit the access to
a bill that the x
well with remember where we do know that the exit would come up here or to use
the one minus six
we figure out what the derivative you would expect x's this negative loans
should be unions
uh... negative index
well we could do here is we could
if you will of the most probable side by making it more immediate negative be
started two years
so we can replace the excellence
negative
premium
in other words we could put that negative on the outside of this integral
i'd
and that's something that we ought to be able to be with us
and of course the internal of using the one-half with respect to you
is
institute for her to do you need to the jury has power
but remember we have a negative that front
and remember what you he is unit is nothing more
then uh... i walked in
one mindset trying to use one nine six
so
we end up with
and negative
two-thirds
pictures
highlands
do you do that for a pet that uses one ninety six
to the three half-hour
in our integrity could negatively to zero
ancestors early in the semester
freelander ninety details later on i would just be able to return to the fall
ac
i think uh...
uh... because i learned in my own calculation here and i i'm sure that you
can come to the same results
greta and what you do that on your own for
great cause and and i'll come back with resort
took after i did my calculations
uh... i got
twenty-eight over three
and remember that's just the area of the less creature
as distasteful
so we still have worked today don't weigh we've got with calculate
uh... the area of the secretary general outreach
so uh... let's go to the next day to do that
okay looking only at the right region right now
uh... america's absolutely concentrate on and see what would be uh...
in of course uh... the graph
is whitewater story below minus six
and here's the
the important part recognize
this paragraph apprentice ha path which is
while one of them is that both the extracts
all that's why he alternatives for a really one-liners x
indelible
today
down below this paragraph
our wine values that are all negative select as defined by white ornamental
serve the warrant six avchd one night at six
slower calculating the
try area and he said to him
we're going to have a a warrior walls all represented for it will all be in
vancouver for
intervals but this is the world
single index in zero
as along here
enormous tom those rectangles an excuse to market right here
so reintegrating principles a little bit exit poll
the highs for representative rectangle he's going to be bllewis on top but nasa
posit why which is where we don't want my sex
in remember we should practice
the launch the walk
all the laws of wind in this case is the woman was out of here
embattled war anything negative for said
yeah yeah
here at the highest live subtract appears below four
and of course forward to that representative rectangle speaking x
once again this is the high end of the rectangle
hinges part is the width of the rectangle
insulin multi those two together we're getting the area a rectangle
one rectangle inward using the interval
content of the year for the number of those
infinitely small with rectangles
in this case
this will simplify it to the internal power
from zero to a woman
earth two times the sort of one ninety six
with respect to x
they are not going to go through this process completely
and would you substitution
day
i mean that i hope that you see
and be as we have
the native of nation we have excellent of big bucks lives where one of my
mistakes were made by sort of land mines that swept through them
so you're tilen
factor the constant pressure to head off
indoor august activated my cynical
insufficiently with many issue nearly one minus sixty-five books on the
previous page
and we do this
we should get
or third
so little area
in israel will be for the people here yet or was it
uh... missus warman bus faces tuning
in a civil war it was truly a theirs
enemies into for a third and fourth there's history in the with thirty two
thirds of which order behalf
concluded that area
i was pretty tough
i want to show you would be easier
and in that's easier way
uh...
the influence of big order
proc
we have the the mickey he's here or what made this difficult
single made this difficult
and assistance to supplement kind of
it yet
isn't even forget the polls
because
when we needed vertical rectangles here
the definition of how we find a highly desirable rectangles
changed
as we move from left to rot
deconcini's foreign x one zero
and uh... it was found in this city
analyst says the excuse for will wreck thing will actually used
horizontal rectangles all we need to do this and i think
there is another internet number of civil
work spanish small in london it should
so was so the next day in the country
goals are all will exchange
uh... and see how it's a problem can be done there
phenomenon ringtone direct
achieve a diagram
while your quality of your call
that we have wife's where d all land mines x
as ria
bravo
and we have originally when i was written
useful ideas the altadena housecleaning
as a straight line this letter was originally written
but i don't like this is a permanent workers with what is this for the story
it may be in the economy is pretty
cso
remember one of eight million figure visited the scale will be concerned
about their selection is pretty good idea
and uh... i
lettuce this for the right up here
is uh... zero on unequal
you all
and of course this is where x is for
fears were expressed cases like today
actually been drawing
vertical rectangles one fine one via instead
brawny
representative for the electrical
criticism though just imagine
these were the rectangles
loneliness
instead of one of the running from world hahaha
so the rectangles are moving in this direction
his lincoln sleep at all levels
weekly calculate be on their indian intend on this
americorps week grozny rectangles
is
website is always on the line in his right side is a little bit grapple
banana tree
so this really don't want to hear
pity
so think about how we would
service interval
so the area
is going to be a reality
the total area is going to be using the integral to add up the
areas of all these little tiny record
but this time
were moving from
lodi
selection gene randall wiring terrific
we're going from below are tied to a high wire
although one using the free
and the highest why valued his father-in-law
so what are your great from the very bottom bonds were thinking of the
rectangles
started down here and women in the stretch
this rectangle has a link that is left awry it has a hiatus
uh... went down
but the up and down on a cd lead that greta small small hyatt in this case and
get a good thing to those who would point out
betsy and a lot of directions as
and solicit having an infinitely small x
dimension isn't even a small wanted ***
and so
with is pty
in prepared in rwanda here pistol-whipped
i said hi ideas but i'm still thinking about two minutes away
okay
yeah at this time the whole though the hike for the length of that rectangle
but we do we take the highest
the largest about you
any
and the largest value is some x values city orthopaedics about you
in the lowest brown williamson extra
they will
here's here through the whole bit this high-stakes value depends on the problem
she hated him parietal
solo cd didn't even need to take this
watch for equal amounts actually needed solvent for expanding edexcel both sides
and subtract what horrible stuff
so they're worried about that horrible
yahoo arrived at ramble about what do you remember
we're looking at the detached excel u
and subtract the smallest x-spam
well the biggest thanks valentino is defined by the arrival
influenza billions to accept it which is why it wants the last word
and also the crash
the smallest expelling
well in in this case
the smallest externally depends on the line and inline
all your online
here's a look at the line which is killari let's close deal
and we thought that perhaps
clueless attraction on both sides in fact solaris system wants to
so that's smallest expect anything more on insult for next year
christian wanted my student
that uh... i think i think you have been there
planning and rewrite the whole country reader
largest expel you subtract smallest air-to-ground ac
and overtaxed and used her loan from our
original
uh... destination those prestigious albeit road for acts
times the
uh... with at rectangle which is part of what you see
we have an integral in terms of why this time it should either acts
there's so much easier and i really don't have to break it into pieces so as
we simplifying their grip and before we try to integrate
well we have all had a negative lots where
and will have a in my list you know i
and we'll have a plus three ac
i've been lost
remember this ends up being the height
this however the olympics of that rectangle if you will
and that he was the hottest won't buy those two together we get the area of
the rectangle
only and this is an easy errol was much easier than what we did although we only
in this case we have cigarettes were originally
and that you can verify this
but will get its act with the same
i would prefer result we have all of you know we added the two different
areas together
thirty two or three
square units will be free to check in
and uh... i encourage you to do that used to make sure that
the chief of the compound
that we've done this for etc
alexis that's interesting and we're going to have in fact employees ideas
remember that we disintegration
were always thinking about
find something that we can define is infinitely small
and whatever work hard to find pretty rapidly infinitely small pieces
he's the internal
juveniles infinitely small pieces together
here so it doesn't matter whether our rectangles are laying down or standing
up
he just changes apparently thought
everyone off on in this uh... section that i'd like to do
and slowly down an ecstatic
who've been our last example about finding area between two crores
uh... we turn on the deals with trigonometric functions highly regarded
by whitewater sinus
and michael changing x where x is between eight a cartoon positive
fabricated
e as i've done it before it gets really critical let me understand what the
classic
and so
that's what's up
was supposed to get ready to
think about these graphs
caseload
uh... normally used to be x's
happening cartoon above the poverty
so will say
this is where they get priority is
will say this is what caused it part of a few years
uh...
soul what we're looking for escaping a few exceptions
now and
it is important to recognize
uh... photograph of water quality division excludes flight can remember
if required a change in the tax
uh... has repeatedly reinstall one particular careers
assaulted we need a power to the prior to that
so retroactively looks something if you will
court clerks
in between these two days made it harder to drop the power to experience
are in fact for experts for their
map
ordered are liable tuned signed x
oracle to sign it
uh... we're coming back and go to the uh... commitment for the silent auction
record
so saludos highs soon
in this little heads native to
and of course
uh...
empire worked really what is the signed a priority
well at school
so why would be three times one or two so
only investors actually ends up being a uh...
nationalization doesn't
so this the side of the options and will collect this
here
in the morning waas
when uh... and certainly a priority
well i would be to have signed it may be part of the two incident made repeatedly
portability is florida's decision
consoles coming there
interest plus
delivery immediately or apologize
something like that
so here's the signed up for
in erie contagion kerr
okay
i was one of the to use a latex
this morning is back
quietly painted cleaners different colours everywhere
f_e_c_ unit we we run into any
problem with the purple pro ste
well one of the problems that we really need to
his early cantilever regions
they were correctly in the area
but bullet point out to you in this case
because we are very comfortable in the surgery
the the cajun function as an aside functioning as
the area in this case
we did say
is aerial was the reaction
and what you think of this is a real war because we do you think of this as area
too
but because of the symmetry
in those jurors saying that if you live in a few times albeit one of those areas
and i'm choosing three times area to because my excel user positive they're
use it just makes things
work a little uses my x_-rays and i'm worried about not so much more about
that
so
we need to be able to calculate the area of the region that we have right here
but now
one of the things that's important here is that we don't know where that would
we though clearly intersected zero
insular were calculated in the area
retrievers more calculating
the area of that second region
we're going to integrate starting as a senior all that's the extent that we
have here
well we don't know what this essay that's
and that's where we started hearing and so we've got to figure out what i did
say it will cover that and more
indeed we do in a all sold to figure out
i think he is a leader the product with the high temperature
and representative rectangular
or a retired tool and i do their that prints were that hochberg picking was
sign of that
and for the highest why value
basically to cite x
that high flyer valued his wise equally is to side effects
handle almost one you belittle me about your defined by the end of corruption
and therefore why is able to tension-filled
says that source we're subtracting exchanging their backs
enforce any
infinitely
stalwarts
is along the x-axis is in the extraction
and so that's the accident scene that we have a renewal of lease ends up in terms
of exit here
but we still don't know all
what this text that is at the intersection or
ansel that's the next thing we need to concern ourselves with
we have come up with that value after
and before we can do this calculation
so let's talk about
how will you find
according to their section
ap
app
remember
overlooking pool
is the extent of you
that produces the same wiretapping
on both tracks
because of graphs have the same accessing martial law in a robotic xd
producers
this is a wine
in both where and that's simply a matter of
saying okay well one of the wisest to sign the graphics
realize injured stack initially it all we're going to be cool
in other words were solving for acts of makes this equation trade
well have been waiting to hear that you can relax change in the next as signs of
the facts about at live co za x
and then we could clear the fraction by modifying on both sides by postal annex
so we don't know where to start a new tax
high school senate x
in little rock is somewhat of a listener exmouth
because we will have uh... multiplied by course acts
blamed one way to solve it ready to make equation is the alternative one-sided
zero on the other situated factor
so melissa drexler naps on both sides
for two three minutes
prostatitis my history it
he all single
and in fact on the left hand side
i have a common factors i next and so it can be factored out
with actors and of itself
concerning the extremist the quantity includes the next month for
people think
satisfactory sign actually both of those terms
well
we have a product that's zero so we take you to the factors
so that i think one of them
this is a person of the concerning explanation nancy congealed
vis-a-vis one of those factors is zero not by the other we will have a zero
that's the way we can get a single missile mopar
well
remember
uh... weekend with a sign of exceeding physio
was sort of ecstasy alters the rule
uh...
when taxes either zero or participant
ornate
so x
can be seen wrote
orson welles wells of party but i remember
were really looking ahead to the within the republican party correctly
so
well bosley alvaro outside our ref
and of course there are a golden exit polls the world's intersection point of
those two questions over a period of aggressive intercepted exit polls we
will
now on the other hand we need to solve the equation to co-sign it's minus one
equal zero
so we are still a good co-signed explaining wanna dividing both
so i'd like to
so we end up with a cosigner axes equal to one half
what where does that have
bolsa x will want half
whenever exteriors
tide you over three
or negative private breakthrough echostar arteries one-half
he goes on a power for his also ok
so there are those are true
values
it is a however three in this terrible expiratory smaller market as yet
inmate arthur is also that it will be cutting are virtually
benso
it with their own business point of intersection has made progress in the r_
three and this one is where x is part of the parlor trick that's really the one
who are interested
because we're just doing the single region in the movement of the role that
area
so ordinary breaking into a positive however three
zero deposited poverty
on our
induced strain our business and we we in their rules
uh...
so what is the investigator grow of to side effects
well its native to co-sign there
logos
probable grabbing her
created to co-sign attacks
in one of the integral of
picture the axe
or
that we don't know it's a little bit
willingness for
well intervening it is the actual also nettles that i've got that
miss it but uh...
well remember the beginning of the semester i told you that there are
certain
uh... enables the g_d_p_ familiar with us and and this is one out
that she needed to recognize
there'd be integral attache genetics
really
punitive reluctant interacts with respective x
is the opposite of the natural law probably after that i could co-sign
effects
now them
that that would be hard for a coming off
point here
that would be too difficult to show
because we can't think of internet access
and sign of x the rabbi co-signed acts
yeah
cutting
and then we could let me do
because i know pets
in that would mean dvds we will come to his first substitution
would be negative sign that connects
the x
solicitor rule will be
assign explorer co-signer satpathy to your family
and we need a scientist the exits that it needs to be negative some of the poor
the negative here veterinarian here
so i would have negative
and in those days and experiences cleanly
in secrecy that's uh... behavioral over you you use ac
actually i think that you really have a negative sa
and of course you tasteless consummate x
pixels
we ended up with
the integral
of the eighty eight you know the x
is
negative
natural absolute value ko sab house
in work in any way from devoting my word for it
well we already had this one dated in there
soul
that is we had the subtraction and configured
historically
needed into uh... simplifying lee atwater native to prove three x
trial from actual are absolute value close friend x
with relatively little power play
for conservative in a car for free
has eliminated two co-signer
proofs on a priority
professional or other action brown you approve something a private reading
subtract
well we didn't below zero
kennewell mobile zero immediately into the central site and see your article
plus natural alternative medical science civil
certainly in
uh...
warning of states
or at least this article scientist
paragraph three is one-half anyway
so they may have to try and save one heck of mister first position
and again we decided that that co-signer paraphrase one-half
and so we had an actual all the rest of the bag and one half which or suggest
won't have to admit along one-half
minus
the eightieth to transform science they go on both sides hero is in fact watch
over-sized egos walkmans
hang
plus the actual autorads that are close friends you know
local fantasy world war i mean it for granted one for withdrawal
remind you that natural
is really
zeros
so
here two years
is really a
made it to judge lance is a native war
could have in this position
plus
natural law that one
*** negative terms they may be treated as a fast food is it
uh...
in the minds of the romans eighteen zero
it would just say closely
so there is nothing more and certainly
ac
uh... warren
one
person actually want half
i think that i've seen answers like this given it a little different way
report out figure anything from the pointed out to be here
natural or a one half of the reverberations natural or what happened
less award of a worship
is so
that's the along with the numerator subtract along with a denominators
and and laura of warranty is the world
so this is the simplifying institution you know uh... it
and once attracts an actual chile
empower people would prefer that
and and that's a proximal e that's approximately you're going through this
whole places
zero point
three and zero seven that we could go through the actual places
insularity area and they were actual