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We've done several videos already
where we're approximating the area under a curve
by breaking up that area into rectangles
and then finding the sum of the areas of those rectangles
as an approximation.
And this was actually the first example
that we looked at where each of the rectangles
had an equal width.
So we equally partitioned the interval
between our two boundaries between a and b.
And the height of the rectangle was the function
evaluated at the left endpoint of each rectangle.
And we wanted to generalize it and write it in sigma notation.
It looked something like this.
And this was one case.
Later on, we looked at a situation
where you define the height by the function
value at the right endpoint or at the midpoint.
And then we even constructed trapezoids.
And these are all particular instances of Riemann sums.
So this right over here is a Riemann sum.
And when people talk about Riemann sums,
they're talking about the more general notion.
You don't have to just do it this way.
You could use trapezoids.
You don't even have to have equally-spaced partitions.
I used equally-spaced partitions because it made things
a little bit conceptually simpler.
And this right here is a picture of the person
that Riemann sums was named after.
This is Bernhard Riemann.
And he made many contributions to mathematics.
But what he is most known for, at least
if you're taking a first-year calculus course,
is the Riemann sum.
And how this is used to define the Riemann integral.
Both Newton and Leibniz had come up
with the idea of the integral when
they had formulated calculus, but the Riemann integral
is kind of the most mainstream formal,
or I would say rigorous, definition
of what an integral is.
So as you could imagine, this is one instance of a Riemann sum.
We have n right over here.
The larger n is, the better an approximation it's going to be.
So his definition of an integral, which
is the actual area under the curve,
or his definition of a definite integral, which
is the actual area under a curve between a and b
is to take this Riemann sum, it doesn't have to be this one,
take any Riemann sum, and take the limit as n approaches
infinity.
So just to be clear, what's happening
when n approaches infinity?
Let me draw another diagram here.
So let's say that's my y-axis.
This is my x-axis.
This is my function.
As n approaches infinity-- so this is a,
this is b-- you're just going to have a ton of rectangles.
You're just going to get a ton of rectangles over there.
And there are going to become better and better
approximations for the actual area.
And the actual area under the curve
is denoted by the integral from a to b of f of x times dx.
And you see where this is coming from
or how these notations are close.
Or at least in my brain, how they're connected.
Delta x was the width for each of these sections.
This right here is delta x.
So that is a delta x.
This is another delta x.
This is another delta x.
A reasonable way to conceptualize what dx is,
or what a differential is, is what delta x
approaches, if it becomes infinitely small.
So you can conceptualize this, and it's not
a very rigorous way of thinking about it,
is an infinitely small-- but not 0-- infinitely small delta
x, is one way that you can conceptualize this.
So once again, as you have your function
times a little small change in delta x.
And you are summing, although you're
summing an infinite number of these things, from a to b.
So I'm going to leave you there just
so that you see the connection.
You know the name for these things.
And once again, this one over here, this
isn't the only Riemann sum.
In fact, this is often called the left Riemann sum
if you're using it with rectangles.
You can do a right Riemann sum.
You could use the midpoint.
You could use a trapezoid.
But if you take the limit of any of those Riemann sums,
as n approaches infinity, then that you
get as a Riemann definition of the integral.
Now so far, we haven't talked about how
to actually evaluate this thing.
This is just a definition right now.
And for that we will do in future videos.