WEBVTT

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You hopefully have a little
intuition now on what a double

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integral is or how we go about
figuring out the volume

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under a surface.

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So let's actually compute it
and I think it'll all become

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a lot more concrete.

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So let's say I have the
surface, z, and it's a

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function of x and y.

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And it equals xy squared.

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It's a surface in
three-dimensional space.

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And I want to know the
volume between this

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surface and the xy-plane.

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And the domain in the xy-plane
that I care about is x is

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greater than or equal to 0,
and less than or equal to 2.

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And y is greater than or
equal to 0, and less

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than or equal to 1.

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Let's see what that looks
like just so we have a

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good visualization of it.

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So I graphed it here and
we can rotate it around.

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This is z equals xy squared.

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This is the bounding box,
right? x goes from 0 to

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2; y goes from 0 to 1.

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We literally want this-- you
could almost view it the

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volume-- well, not almost.

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Exactly view it as the
volume under this surface.

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Between this surface, the top
surface, and the xy-plane.

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And I'll rotate it around so
you can get a little bit better

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sense of the actual volume.

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Let me rotate.

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Now I should use the
mouse for this.

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So it's this space,
underneath here.

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It's like a makeshift
shelter or something.

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I could rotate it a little bit.

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Whatever's under this,
between the two surfaces--

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that's the volume.

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Whoops, I've flipped it.

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There you go.

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So that's the volume
that we care about.

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Let's figure out how to do and
we'll try to gather a little

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bit of the intuition
as we go along.

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So I'm going to draw a not as
impressive version of that

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graph, but I think it'll
do the job for now.

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Let me draw my axes.

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That's my x-axis, that's my
y-axis, and that's my z-axis.

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x, y, z.

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x is going from 0 to 2.

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Let's say that's 2.

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y is going from 0 to 1.

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So we're taking the volume
above this rectangle

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in the xy-plane.

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And then the surface, I'm going
to try my best to draw it.

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I'll draw it in a
different color.

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I'm looking at the picture.

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At this end it looks
something like this.

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And then it has a
straight line.

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Let's see if I can draw this
surface going down like that.

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And then if I was really
good I could shade it.

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It looks something like this.

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If I were to shade it,
the surface looks

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something like that.

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And this right here
is above this.

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This is the bottom left corner,
you can almost view it.

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So let me just say the yellow
is the top of the surface.

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That's the top of the surface.

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And then this is
under the surface.

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So we care about this
volume underneath here.

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Let me show you the
actual surface.

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So this volume underneath here.

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I think you get the idea.

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So how do we do that?

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Well, in the last example we
said, well, let's pick an

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arbitrary y and for that
y, let's figure out the

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area under the curve.

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So if we fix some y-- when you
actually do the problem, you

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don't have to think about it in
this much detail, but I want

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to give you the intuition.

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So if we pick just an
arbitrary y here.

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So on that y, we could think of
it-- if we have a fixed y, then

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the function of x and y you can
almost view it as a function

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of just x for that given y.

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And so, we're kind of figuring
out the value of this, of

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the area under this curve.

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You should view this as kind of
an up down curve for a given y.

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So if we know a y we can figure
out then-- for example, if y

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was 5, this function would
become z equals 25x.

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And then that's easy to
figure out the value

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of the curve under.

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So we'll make the value under
the curve as a function of y.

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We'll pretend like
it's just a constant.

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So let's do that.

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So if we have a dx
that's our change in x.

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And then our height of each of
our rectangles is going to be a

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function-- it's going to be z.

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The height is z, which is
a function of x and y.

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So we can take the integral.

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So the area of each of these is
going to be our function, xy

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squared-- I'll do it here
because I'll run out of space.

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xy squared times the
width, which is dx.

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And if we want the area of this
slice for a given y, we just

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integrate along the x-axis.

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We're going to integrate
from x is equal to 0

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to x is equal to 2.

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From x is equal to 0 to 2.

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Fair enough.

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Now, but we just don't want to
figure out the area under the

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curve at one slice, for one
particular y, we want to

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figure out the entire
area of the curve.

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So what we do is
we say, OK, fine.

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The area under the curve, not
the surface-- under this curve

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for a particular y,
is this expression.

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Well, what if I gave it
a little bit of depth?

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If I multiplied this area times
dy then it would give me a

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little bit of depth, right?

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We'd kind of have a
three-dimensional slice of the

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volume that we care about.

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I know it's hard to imagine.

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Let me bring that here.

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So if I had a slice here, we
just figured out the area of a

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slice and then I'm multiplying
it by dy to give it a

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little bit of depth.

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So you multiply it by dy to
give it a little bit of depth,

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and then if we want the entire
volume under the curve we add

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all the dy's together, take the
infinite sum of these

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infinitely small
volumes really now.

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And so we will integrate
from y is equal to 0

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to y is equal to 1.

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I know this graph is a little
hard to understand, but you

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might want to re-watch
the first video.

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I had a slightly easier
to understand surface.

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So now, how do we
evaluate this?

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Well, like we said,
you evaluate from the

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inside and go outward.

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It's taking a partial
derivative in reverse.

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So we're integrating here with
respect to x, so we can treat

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y just like a constant.

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Like it's like the number
5 or something like that.

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So it really doesn't
change the integral.

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So what's the antiderivative
of xy squared?

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Well, the antiderivative of
xy squared-- I want to make

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sure I'm color consistent.

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Well, the antiderivative
of x is x to the 1/2--

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sorry. x squared over 2.

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And then y squared is
just a constant, right?

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And then we don't have to
worry about plus c since

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this is a definite integral.

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And we're going to
evaluate that at 2 and 0.

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And then we still have
the outside integral

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with respect to y.

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So once we figure that out
we're going to integrate it

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from 0 to 1 with respect to dy.

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Now what does this evaluate?

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We put a 2 in here.

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If you put a 2 in there
you get 2 squared over 2.

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That's just 4 over 2.

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So it's 2 y squared.

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Minus 0 squared over
2 times y squared.

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Well, that's just
going to be 0.

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So it's minus 0.

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I won't write that down because
hopefully that's a little

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bit of second nature to you.

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We just evaluated this
at the 2 endpoints and

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I'm short for space.

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So this evaluated at 2y
squared and now we evaluate

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the outside integral.

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0, 1 dy.

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And this is an important
thing to realize.

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When we evaluated this
inside integral, remember

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what we were doing?

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We were trying to figure out
for a given y, what the

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area of this surface was.

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Well, not this surface, the
area under the surface

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for a given y.

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For a given y that surface
kind of turns into a curve.

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And we tried to figure out
the area under that curve

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in the traditional sense.

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This ended up being
a function of y.

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And that makes sense because
depending on which y we pick

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we're going to get a
different area here.

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Obviously, depending on which y
we pick, the area-- kind of a

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wall dropped straight down--
that area's going to change.

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So we got a function of y when
we evaluated this and now we

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just integrate with respect to
y and this is just plain old

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vanilla definite integration.

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What's the antiderivative
of 2y squared?

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Well, that equals 2 times
y to the third over 3,

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or 2/3 y to the third.

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We're going to evaluate
that at 1 and 0, which

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is equal to-- let's see.

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1 to the third times 2/3.

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That's 2/3.

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Minus 0 to the third times 2/3.

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Well, that's just 0.

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So it equals 2/3.

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If our units were meters
these would be 2/3 meters

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cubed or cubic meters.

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But there you go.

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That's how you evaluate
a double integral.

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There really isn't
a new skill here.

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You just have to make sure to
keep track of the variables.

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Treat them constant.

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They need to be treated
constant, and then treat them

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as a variable of integration
when it's appropriate.

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Anyway, I will see you
in the next video.

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