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well we know that polar coordinates are always useful when circles around the
origins are involved and that's exactly
we're poor courts are going to help us with government let's get to it
uh... basically
when you've got rectangular coordinates and also very inconvenient for marketing
disks because after all we would need functions by equals plus minus
screwed a script minus six square to model the
circle that gives the boundary
all right now in public awareness of the audience here
because we would just let artie
polar variable are run from zero to the radius of the disk
and they would let fata around from georgia to client that would cover the
whole describe
so v
questions that we need to answer before we can actually go to integrationist
what the every element dat looks like in polar coordinates
and at that is basically because previously
well sure i think we have rectangular area elements of the d_a_ was d_x_ ki y
and that's all for being a c_n_n_ worked
so that that was not a problem
but uh... win that we are going to curb coordinate systems all of the seventeen
point
technically have to prove a
for being a figuring
that works for the polar coordinate system and for any other kind of crap
coordinate system
in cell
physical analog off that is to just determined with the area and to
determine what the area and it looks like
the
formal transformation theory is something for a
separate presentation on coordinate transformations in
multiple integrals
here we're going to take the geometric approach
which is the approach that you often find in
physics as well as an engineering and it works perfectly fine it gives the right
result
because if we're looking at gary elements and parking in coordinates will
basically
we would look at an ex white women system
and we know that the coordinate lines are the horizontal lines why equals
constant as well as the vertical lines x people's constant
instilled in our area elements are little step in the x_-two richards lead
d_x_ times will step in the white *** d blindness are
d_a_ for double integral in fact it's like
uh... will be a steering works
now if we're looking at that impala record notes we have to remember that
uh...
z z coordinate
line support occurs and paula quarantine the worst eleven x y coordinate system
typically overlay are
that we're looking at are equals constant which are constantly circles
a little bit lines *** or at curves for state is constantly charge these
worries
off the aldrich but we can see therefore here yes
that the naturally area element and polar coordinates actually is something
that is bounded by to resolve the origin
infineon
segments
off concentric circles around the origin
one of the things that we can already realizes that if we
make this partition fine enough this pool sort of look like a rectangle but
we still need to figure out
with the links off the sites are
in so basically the poor rectangles
and they literally are caught poor rectangles equivalent to look at
rectangles
quote unquote that are bounded by an inner circle of radio stay an overseer
of radius b
if then they are bounded by a rate that comes off the origin angle althought
in another rate that comes off the origin at an angle beta in their
directing the will be
exactly this and then of course for integration we have to partition that we
do that
with more concentric circles
as well as with more
resolve the origin and again we get these
tiny corner rectangles
that are
area elements
all right so what's
approximate area off a
tiny corner rectangle well we
grassley tiny corner rectangle it starts at a point
are fed up and then just goes in eight dismissed p_r_ and so certainly one of
them in jesus just going to be
the difference in the radii which is ultimately going to be a differential in
d_r_
then we draw the segment also
concentric circle in the game going ciara drove a segment of the other
concentric
in opera help that person tells us of course it as we go into all the way
our caller rectangle starts and angle fata indent insanity angle theta class
eighty little
do you think that a little distance
inflate a
and so begin
we magnifies that barnett it will make the delighted that defense more than i
think this really is approximately rectangle
so i think the good side links will be d_r_ int the rates i think it will be
just
a little lanes of the circle that that has been
cut out of the circle by this angle
defeated insult that would be
allergies data
uh... which case do yes and that's again just from the definition off afraid you
measure actually
okay so that means
the area off a
differential corner rectangle is dat equals archaeology theta
and that means azathioprine
if f_ of x_ is going to go over the poor region are
where the border region are is defined by
uh... eight yes eric larsen it'll be what we want to ask eight-b_ credit
report and zero
impact they had the baby pandas alpha and beta
indian political bra raised all integral
of if over the rectangle are against the generic area element bs
into the hotel for two data indigo for eight a b_ those are the balance in our
infant doesn't that make sense
and then we have if of course now that if is evaluated the polar coordinates
it's no longer ethel extent clients
if off
are co-signed theta alpha are signs and of course that's what ex-employee
hardcore coordinates as we recall
isn't this part here is the
polar erie elements of this list
basically now something will be go back to this
intuitive ideas that
double integral is basically a continuance some off values off the
function multiplied
with little areas it's just that now the new areas
have value r_j_r_ d_c_ that formula
works beautifully in also
uh... for those of us who are concerned about the formality behind it
care real justifies its
formula also once we have
general
abstract transformation for me to ask for double integrals
in for triple integrals it's just a as i've mentioned before
integration actually use
also formally very demanding and so
this geometric approach because it gets so quick results really is
preferable over going through i don't know
is years' worth or two of graduate level mathematics to fully formally
justifies
okay so
ought to be working with examples one it's going to quit the function info xli
equals
two-minus square root expert class was great over that half age off the disco
reviews too
that isn't the first and second quarter so we can see already
that that's partly because
if i had to integrate square it affects their class white square
uh... using the ex ante wife that something that
i'm not sure even is is possible in principle
uh... but let's take a look at what the domain looks like well so we
get ourselves index like warning system we're looking at the half of the gifts
***
that is in the first and second quarter and so we need abounding circles of only
two radius two
into the race to its hearings
then z extended native correction is of course negative to and they have both of
these kids now
mark here
all case so
how do we integrate a function over this case will i don't want to do this in
rectangular quart tried
because when it comes to do support accordance are simply unlocked more
appropriate so i would have the radio school from zero to to uh... would have
the angle go from zero
took pride in that means
if i make my teeth in an hour ago i would start with integral zero to apply
integral zero two two four the radius
two-minus square it excluded plus my spirit that would be to minus
screwed on straight
is that i have the area of the man which now is r_j_r_ g st dot
in i've got the rest is just competition actually the computation is
surprisingly simple because what i did i get to minus our times are so that would
be to our minds are square
so i have the same into grozny off
to our minds are straight can you integrate to our minds are split with
respect to off course you can
we get into riveted our script minus one third are cute
and so we have to be closer to twenty they tested on the outside our script
minus one third are cute from zero to too
by with leading zeros just get zero so we only have to worry about the upper
bounds we get
mindlessly third swell for its twelve thirty one st there it's it's four third
so we
and that would be one of four months haters minus zero one zero actually here
so that would be the into from zurich a pile of four thirds and that's of course
just for the first fight
uh... because predicting a constant
in fact it quickly
if we had said up these integral
in rectangular coordinates basic implicitly what it looks oracle
that's ob basically we probably wouldn't have even wants to start so i think that
would have been in the people that we would have to grind through no merit
that being said it's still nice to be able to solve such integrals
precisely because at the end of this presentation or going to derive
precise value of the into going off the standard normal distribution across the
whole realigned which is something that we have
postponed from single vehicle calculus
right now let's just take a look at a few more things for example
i may want to integrate the functioning for ex-wife equals y over the region in
the third quarter dies down a path the circles
it's great plus widescreen equals forty expect us what's critical sixty
well we've got to see the cell
we draw
we give s_l_s annex like awarded system we draw a circle
of radios to which would be the circle expect less less greedy equals four
we draw a circle of radius for which would be the circle expect us what's
critical sixteen
with interest in what happens inter
first quadrants of the region would be
in the third quarter and here which means it would go from fata equals tiny
to later equals three point over too
this access to this access and with go
therefore
from staples by tuesday april stricken over to what was going on the radio
salute gopac
and we would call on the radius one more time so this is
the regions that we want to be degrade over
which means
that again once we've got the picture of the rest is really just computation
because
if we want to integrate the function f over this region against the polar area
i mean well what do we have we have
stated goes from
pint is free play over too
argos from two oh two four
the function is wide well my views are signed fayda
in the area amanda's r_j_r_
do you think about well
i think on the next one just simplifies it into our spirit signs they argue i
was just saying that the colonial
r squared science data and
well no i'm going to integrate
r squared
and that would mean i get a one-third are cute
signs they are that he doesn't do anything i could have also factor is out
in computers
is a product of two separate integrals but here you get one third are cuban if
you set up like this you just have to remind yourself that articles from two
to four and we don't substituting for friday yet
well one-third are cubed okay so we would get act four we would get four
cuban which is sixty-four so sixty four thirty minute speakers here
so we end up with the same integral
sixty four thirty one is the first time so i think that the fate of
sixty four thirty one speaker it's it's fifty six thirty so we end up with fifty
six thirty
times the injuring two books on fri which is made of was something out
going from pune to three pile ritter win here's what we have to count the
negative signs
beacause
well the course of three title virtue he's zero
that was signed off by its negative one but that is
subtracted to read it
minus
negative negatives one that's three native sons to keep limited signed over
so we have made a fifty six thirty if that's the result does that make sense
well actually it does
because when we integrate y
over region where y is negative and that's what we're doing here is got to
get the negative result
so this actually at least the sign is correct and the numbers
should also have worked out
correctly here as if these are the kinds of
sanity checks that you also want to just execute for yourself
to make sure that these there are a any gross
errors in your computation of these are also the kinds of things
that you can ask yourself
ukraine seems through any numerical computation because at the end
you still want to make sure whether you have sent integral correctly and that
sometimes you just
push things the wrong way with the angel infectivity is in effect
the sign for it
***
well what else do we have
i want you to compute this physical here fantastic
uh... that's exactly the kind of integrals that a couple of panel circle
i told you we would want to avoid only solved numerically
because after all i don't know how to integrate the square root of one plastic
script oswestry begins jeewa height
is quite horrible at least
at yankee they're they're probably art
ways to do this with the integral table if you factor out the one 'cause export
but rooms this is goodness graciousness
this as giving back
insole rather than working this out and rectangular coordinates
that's just robot the region books quell
the region goes
from negative four to four and reintegrating over to semi circle here
because after all
outward
why goes from zero to sort of sixteen ninety-six whitman idiocy something like
that
you should have the reflex still from calc is a pre calculus that's a sense
n_c_r_ percent certain people from eighty four zero
so it's a very fast off the upper semi circle
so actually what happens is that reintegrate over a
speed is
over a quarter disco craze for we go from negative for
two zero white to the function is
sixteen
minus x square
into we're integrating basically just index from negative four zero invent
from white to
from zero to
sixteen months x squared
and us so that is
these quantities paper integrating offer i think
we're not going to have a wide one we just have this white rule here because i
think recycling slides
uh... because most certainly these things take a lot of make right
so that's what we want to get a great over and that means that these integral
in park warden that's
is an integral that goes in fatal from pune over to two pi indian are from zero
to four so we end up with an integral from pilot to the party
in the room from zero to four in the radius
score of one plastic square class whites criticism layer of one-plus r squared
arterial images archaeology theta as before it now
we can solve this with a quick substitution req
because
if i do a substitution due week walls
one class are squared
world indeed eulogy artists who are beach
works out in the substitution exactly such a way that these are here cancels
the are denominator
ultimately
the rest is exactly just this kind of computation if
that such as usual is something that you don't want to do mentally i don't blame
you
i would you do dude with paper and pencil i'm just going to show you
result here and that he is
the integrity of this one third one class arts critic for the three hasn't
if you see something like that in your book
at least as you read along to completely verified that that works out because if
i take the derivative here
i'd end up with
three hats
times one-third which is three hats
times one-plus arts greatly three has minds what one which is exactly the
spirit of montrose arts critic
time said review of the inside which is to our
which together with the one hand we still had left
gives us
the artistic kind of mental calculation that ultimately if you can do it
in this kind of stuff will not bother you at all
if not
there's no penalty for using
paper and pencil just a few i can also
uh... by not putting all the stuff you know i can put it on one panel and quite
honestly
infected sticks
people are going to put it in the so i want you to get used to doing this
mentally or with paper and pencil in the margin twenty px some separate notes
that you take
already end of certain that we have to keep going here
at the upper bound is forty is forcibly would end up with
fourth-grade sixteen plus one seventeen to three has to one-third seventeen to
three has in mind is at the lower ground this time
we don't get zero we get
one-two-three gas which is one so actually what we have this at b keep the
integral from pi testify
seventeen to the three has minus one define by three
in this league in the great that the state of well that's just
another multiplication with pi over to a subdivision but you would give us the
stakes here
and the result ends up being seventeen to the three r's minus one
divided by six
in so what we can see here again is even though
it's just computation and i'm certainly still playing the task master here
and i'm not going to college i saw that i wanted to be on top of these
computations this is the kind of stuff
that we have played with enough in single variable calculus and so now
it is just time to step up
in fact you get up in knots jazz dance on it t_v_ character also off the set
and uh...
yes other than that
week and i think also agree that the integral in studying at first but by the
time we've reached this
we breathe a sigh of relief because this looks like something we can tackle and
that goes back then also
to the mantra that i certainly repeating at the impact this part
off the course also incentives
the beauty is no longer impeding beautiful results as in the answer is to
work the answer is four
the beauty of this stuff now lies in the very fact
that we can get results and all that you know that if the results look like this
alright removal unencumbered
also take a look at
what this does in triple integrals well phone records of course don't give a key
connected triple integrals but
we can
encode eight region in dublin to in double integrals i think that your legal
s
accor reported for its intentions
simply go in the third dimension so for example
if i want to compute the center of mass of the object to the right of the x_
claimed that is going to be no by the column
z_ equals crew expects way straight and up after the plane
z_ equals to one of the first thing we have to do is we have to visualize
we know what it called looks like we know what a truncated cone looks like
that would be
something like like a spinning top that u
c asp net may have seen as a child stoic and if we're looking at something that
is to the right of the xvi playing then we just have to cut this thing
politically
so we have the stuff that is to the right of the ecstasy plane it insanity
equals two was so this is pretty much the architect that looks like
uh... that
if the object is described here
in idea on how we can see of course that that this is a make up a little
coordinates antes de x and y dimensions
because of which ethnic our state and local from jericho height
and would lead a rally is cool from zero to two infancy would actually go test
from
spirit expect us was pretty screwed up our spirit so that's actually pretty
good to know
so as you would go from our
so that means that the integral
other than the total mass of this thing and ultimately the center of mass for
mother but also review
well the total mass would be the triple a_ posted in cd
dvd
well whenever we've got that we're going to assume you that this thing has
constant density of roe and that's going to show
that the density of ultimately going to
cancel out so even if we don't have an explicit density given
typically when that is the case we assume that
object has caused the density cause otherwise we would have told you
how the density berries
we just have to track that through well no problem
in together for me that we need to recall
is
the formula for the center of mass which is going over in times
the triple integral off the generic position vector x_y_z_ off eight point
inside the object
begins the differential mass element which is
density times
the differential volume and again we're just assuming that this after test
constant density of the rest
because we already see how he said up the integral dresses
quote-unquote just computation let's take a look at
for the next
we get that the masses of traveling across the density
against the volume elements of that would be that
are angle went from zero to apply
operators goes from zero to two and then we had
feds eagles from screwed up the radius squared so
as he goes from the radius
isn't the density is just
very well its role in then we have dec
is for the calendar on the outside the uk
the areas and
are d_r_ do you think that well that is something that we can certainly worked
out
rather nicely in again the rest now really is just computation
because the factor of the role to get the doubles your client mls zero two two
integral radius
to off the radius dvd or do you think about
well integrating pickens these the
will give us
two-minus our times are right
ten so that would be
to our
minus r squared because
was the artist consonant so we just integrate
uh... once a week s
c from are too which gives us two one song and dance just multiply out to give
us this with the are now we have to integrate
d_r_ well that would give us
i square it minus one third r q grant
so yes we keep thoroughly keep the outer integral our script minus one third are
cute from zero
too too which gives us for
minus eight third swell forest well thers minus eat fruits it's for third
three evro integral zero two point four thurs dot d
they are so that would give us for the first pyro
because after all integrating the constant just
gives us a multiplication with that
difference of theo
between the bounced so performance or about
and so the masses
four-thirty highbrow and so now the king for the export nets
the exco and it is one over in times of trouble in the below the arctic x times
wrote dvd
which is one of the mass
time genitals are applying a bills routed to a nickel are too
are porcine state at times the density
jeezy
archaeology say dot
we're setting that up
and now we just have to
ke computing ever going to keep the one over in out front because
that literally just
clutter up the picture alright the rest is just computation
so we have one over and we put all of the rope
again have an r squared we've put was something out
easy d_o_d_ theta okie reintegrate
includes dec
and so that we can give us a two-minus are
which gives us that the most of the two months are with the arts crickets as to
our script minus r q
we keep the course on data we've got the integral jerica too
uh... the art deefaye dot
is that gives us wondering times row interval zero to apply
well to our spirit gives us an andy griffith two-thirds are cute
are cubans naked directive minus one sir are for the fourth costly minus sign
here
from zero to two that would give us
uh... sixteen thirds minus
minus four
which would be sixteen thirty minus twelve turns which would be four thirty
p_m_
so we have one over in row integral zero five four thurs was signed fata india
from zero to five was signed fate at the theater
okay so we should have seen about one without the integral
uh... even though the indigo wasn't that bad and of course i i set that up
deliberately just show you that
kit and downstate
york at what the computations meeting
we should have seen this
without the integral because if we're looking at an object that has constant
density
that is symmetric with respect to the why is he claimed and of course the
integral
index is going to be symmetric
no matter how off middle cursing the tripling the comment that means
the export it
uh... must be
zero physically speaking but also makes sense these optic well-balanced
along any taxes at least indexing for a balanced below
any acts is that we put forward
that is is going through with the
why easy planet so that means
the x_ cordoned off the center of mass for the must
zero here
unfortunately the other ones are going to be so easy cause even see aswell yes
in the uae *** we don't have
symmetry
in so quick note to yourself not to ourselves
when the disney is constant
let's look first century that is something that can always tell
especially for jets
that have constant density
but now
well so we'll do that from here on both of these optic it's not gonna work so
we have to bite the bullet and actually compute the white when it the hard way
cause there is no other way yet
so like what it is one of her in trouble in of all flight times ro
dvd
which is not over in which we're going to drive through the computation
integral zero to apply integral zero two two indica from are too
our
signs data which is why
times ro
dec
r_d_ argues that which is
the volume element we have that the extent as well as the
dat for the integral
in the x
weapon apparently ex-wife and again the rest
is just can't get a chance to rewrite that up
pull out the role we keep the integral
we could make the arent this are into a harsh critic that science data we've got
dvd r t say dot
well the integral
against easy because r squared
in signs they are constant settled just give us two-minus are as an additional
factor which
then of course when they multiply the two-minus our v_c_r_ spec is just
watched great minds are cute
everything else stayed the same right one over in a row
is one of the in wrote the article is still the same with synthesizer
we still have the d_r_ we said to be safe
all right you know we have to integrate in
d_r_ that would give us two-thirds are cube minus one
fourth
are too is a
forth like we
also had previously
so that gives us a lot of rim times role from here the outer integral
zero to apply debate of states
the same way right
is uh... then we
simply have the two-thirds are cute minus one for the heart of the four th
which is exactly the integer default port because here
et zero we get zero act to we get sixteen thirty minus four which ticket
four thirty
so this time to end up with one over in a row integral zero two five four thurs
sign in state at the state of right now of course you have a printer life
science data is not zero
so we end up
uh... gap or piece of the integral
zero to apply off sign say now actually
is too and that's because the into derivative is the negative was unfair
acti
gives us one
n dash zero gives us minus minus one one minus megan one page two
so that means begin
eight thirds
and the one over in because it's for third stones to it and we have wondering
and roskilde
it now we have to planning what in is flint was for a pyro define by three so
that
went over in this three or four pyro times eight thirds ro
the three e's cinsel
the age of the four kansas saw in the rose cancel so we get
overpower here is the answer
and that certainly also makes sense because an object of constant density
for an optical coincidence city the coordinates off
the center of mass can only get to know that he on the tree they can't depend on
what the constant for the density
and that's approximately point six three six six which also makes sense because
if we're looking at this
off a cold that we have here
most off the masses concentrated new jersey axes and that means
that's the wide cordoned off the center of mass
must be closer to two than it is
zero
okay now i say it's keeps us that's unique in milk
just computation in fact indian face-to-face classes i would probably
often just
skip the states and just
i give people the answer because it takes quite a while to do the steps
because we simply
would want to cover more examples we would want to cover more stuff where we
do the interesting stuff mainly the modeling in these presentations i've got
a little bit more liberty because we can do these computations
quickly because as you have already seen hopefully not too you should bring
these competitions going off faster
that face-to-face class because i've got to run my own view to use the pause
button wherever there is something we unique defendant gets an again this
remainder
reminder for the next one for that
z coordinate
use the pause button as necessary here we go
physique orders but over the mass times the trouble in the goal of zero
z times wrote not zero
zero would be really nice but debt speaks not in the socks for us
so that would mean we have one over in integral zero to apply integral zero two
two for that
uh... for that half of the disk integral hearted to which gives us the
integration of the cold
of z times wrote
d_c_ invent hardy our teeth data is
the area of the mint
contingents
and off
the races again just computation eureka
whatever in religious fought off the road this time around
uh... the combination is really simple interest
pulled the are enough to have should have passed it on the outside but either
way
the editor of the faults the as one hats the squared
which has to be taken from harder to
all the outside sentence a couple of weeks he stays the same celebrity also
practicing
reading skill here which is we just focus on the stuff that
changes
in dire well one hence the square at the upper bound gives us for over two which
is to win at the lower bound
it gives us
one-half
i swear to end up with two minus one-half r squared
in a tunnel applied that helps
with the are on the outside i get to our minus one-half or q
how exciting it will stay the same i think that one of a in a row got run
over en route with the interval zero to apply in a browser in the two and a
blizzard of particles in order to
name we've got to do you already set up which is also here
integrating that begins
art would give us
r squared
minus
one
eight art to the fourth right
and so
that's that again outside stays the same we keep the row indian we keep the
outside integral as it was us great minus one eight art to the fourth act
to gives us
for
minus
well tuned to the fourth is sixteen so sixteen array to four months to gives us
too
indict at the lower bound picketers zeros so we
the role of the and we keep the article in the in between is to which of course
just gives us a
tomb pride and so we end up with one over in to apply
times
rogue is now we know
that the mass was four pyro
over three so wondering as three or four pyro beekeeper to apply to keep the rope
we realize that the role cancels as it should we realized that the pie cancels
that doesn't necessarily always have to happen but it does happen here
is then we have that the two eight games before
cinsel so we had the president on a tour
too and we end up with an answer
handouts that you can also makes sense because most of the mass
office
off the coating is actually music talk
if most of the masses near the top of the center of mass ought to be
closer to the top which is to have been to the bottom
which is
zero ok so the center of mass
it's zero
to over pipe
three house and again that makes sense
it makes sense
because
it makes sense because
here's a geometric optic right
symmetric
across
why is he playing which means for constant density
aches coordinate
all for the center of mass has got to be zero
uh... incident
most of the mass
near the
x declined and that means
that the likelihood of the center of mass repeat
closer to the left end then the right in ended is because it's point six
something which is close to zero
that the two
is most of the mass
new york
the upper end of the object which is to work not in years and over in which is
that you know what that means
three has z coordinate also makes sense
for this sandra hodge now of course
we still could attack computational errors that are a bit more settled at
least but it is we have these
sanity checks when we are connecting
integral in the computations to the physical reality and this is a result
that make sense as we started out too
reality
brings us to
another part off
physical reality which gives the central distribution quiere
uh... single bearable calculus b postponed a few things because
couldn't compute the intercollegiate anita exclude over to *** exp
classified
and brief we said that repeatedly that it's a function that doesn't happen in
future every student derivative in the nineties
codes
form
all right
what can we do here well if i squint at this thing
i might as well bike that is the end of amazing ability to infinity each of the
native exclude over to d_x_
time see if we both make a difference in acute from a conflict between finity
euthanized wide spread over two
delight because there is no rule that says that he had collate dummy variable
extend went in to grow and why in another article
but of course no i can't pull this together and don't want to send this
becomes a double integral
don't worry about the minus infinity to infinity that just means
with integrating over the whole space so technically
we're looking at and improper dublin because we're not going to go through
the technicalities for that
is suing that works out
and it does
we integrate
uh... now into the native x_ whatever two times you do negative ways went over
to you
and of course
when you multiple exponential ski at the expo dozens of the exponent on person to
speak to the negative expect us way spread over two
delight d_x_
in haiti it's great crossways great is our screen
into the area element in colour quarters gives us an extra are so weak and was
substitution here but
so this it's the dotted to go over
uh... the whole two dimensional space in determining if x
spectrum's west went over to dat and if i can just
massage my d_n_a_ in such a way that it's going to speak or something
that gives me a good substitution and that's what polar coordinates do
i off to the chance to computers
we are certainly hiding as a separate we're hiding against certain technical
aspects that they are best left
for advanced calculus class or even three beginning graduate level class
uh... in mathematics
we're just going to see you that as long as things work out we will get the right
number
and that's not bad either considering that
we just started working with calculus a little bit more
than a year ago ok so
if we want to compute the into political ex-wife equals ito negative expect
oswestry over too
over a discovery his a center at the origin
that's something that that's needed it
that is something we can do sorry for the star in it
uh...
if we can take the end of the integrals estimates goes to infinity than that
basically is the into growth dysfunction
over the whole space
invest in something that we can cause look hard
take the inner core over this disk
of each of the native expect us what is going over to
d_n_a_
well that'll be the indigo from zero to the radius
beacause we have to take
if we're going to get the whole describe happening are run from zero to the
radius
cut allied in the inner envelope will be from zero to two pipe as we have to go
around the world is skewed data
my ear here element is r_j_r_ do you think i had my function
has simplified malfunctions each has a negative are screwed over to right and i
think
that we've talked about every so often is even though it feels like we're
adding insult to injury after we've done all this stuff in multiple variables
the next thing that we always hunt forty s
that we try to get rid of some of the variables and if i can turn it back
square persuasively
into a nondescript
where it is
in integrals i'm talking about or whether it's a partial differential
equation i will always want to take that because
the fewer variables i have
the e_c_u_ the computation is
and here is that of course i can puzzle this apart
i have an integral from zero to eight
d_r_ of the country to negative our spirit
uh... times are
into the rest of my degree it is just the ones or just a minute from turkey to
pi
detail which gives me a two point
directly social work this out
here is of course where
i again want you to use paper and pencil this is a quick substitution the
interests are from zero to two point do you think that that's just
too high
that's not a substitution that's a big tax
uh... but what we have here is of course if they do a substitution you equals
negative are square over to that everything works out in such a way that
ultimately kit
into derivatives here
because you're going to see this stuff invokes the way you read this is
used to be verified that we're talking affected cigarette if you're one of the
derivative of the exponential functions the exponential functions self
so i get you to native are spread over two back
and i have to take the derivative of the exponent which is negative
to our over too
with this negative sign that it just gives you these are
indicate
with paper and pencil work it out of the substitution is you equals negative are
screwed over too
and that should be
feel the heat you need
to solve
all right
we move on reno pledging
balance
insulin allen articles a day i can't make it a few dozen if he's going over
to a are equals zero a kitten
gate into the zero with witham which was a minus minus is actually a plus one
in so that means this is to apply
one bite issue to the negative he screwed over too
in fact is
the integral
off of the function over that over these disco radio station to get the origin so
the first part
solved here
i know it i think that it is a goes to infinity wanniski goes to infinity
because he doesn't need it he screwed over two groups as your about is too
if that means
this the information from the preceding panel
that means that the interval from the defense in the key to infinity of each
of the negative exclude over to d_x_ quantities queen
to hike
and that means
that the indo for megan's affinity to infinity
of eaters anita fixed-rate over too
d_x_
squared off to buy
just press as we came to all along in the statistics portion
off the single variable part off this course so this is something as you can
also hear my voice this is something that
math people are quite enthusiastic about
in fact is
because this gives us
the exact skating value that we need
for it
the standard normal distribution which is
certainly nuts
you have seen the examples of just a body anything that you can do
polar coordinate so now it's time for you to look at some of the examples
off integrals in
rectangular coordinate that
coordinates that we have to rewrite
were also at some
integrations we simply are into creating overstuffed that has uh... that that is
circular in shape or anger in shape
two concentric circles and the stuff in between
or you can
parts of these things that are sitting
inferior use
uh... quadrants money as you can see
i'm having fun with it i'd hope you will have to i'm going to take a break