Riemann sums and integrals

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.