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## Multivariable calculus

### Unit 4: Lesson 5

Double integrals# Double integrals 6

Let's evaluate the double integrals with y=x^2 as one of the boundaries. Created by Sal Khan.

## Video transcript

Welcome back. In the last video we were just
figuring out the volume under the surface, and we had set
up these integral bounds. So let's see how to
evaluate it now. And look at this. I actually realized that I can
scroll things, which is quite useful because now I have
a lot more board space. So how do we evaluate
this integral? Well, the first integral I'm
integrating with respect to x. Right? I'm adding up the
little x sums. So I'm forming this
rectangle right here. Right? Or you could kind of view it
I'm holding y constant and integrating along the x-axis. I should switch colors. So what's the antiderivative of
x y squared with respect to x? Well it's just x
squared over 2. And then I have the y
squared-- that's just a constant-- all over 2. And I'm going to evaluate that
from x is equal to 1 to x is equal to the square root of y,
which you might be daunted by. But you'll see that it's
actually not that bad once you evaluate them. And then let me draw the
outside of the integral. This is y is equal to
0 to y is equal 1. dy. Now, if x is equal to 1
this expression becomes y squared over 2. Right? y squared over 2, minus-- now
if x is equal to square root of y, what does this
expression become? If x is equal to the
square root of y, then x squared is just y. And then y times y squared
is y to the third. Right? So it's y to the third over 3. Fair enough. And now I take the integral
with respect to y. So now I sum up all of these
rectangles in the y direction. 0, 1. This is with respect to y. And that's cool, right? Because when you take the first
integral with respect to x you end up with a function of y
anyway, so you might as well have your bounds as
functions of y's. It really doesn't make
it any more difficult. But anyway, back
to the problem. What is the antiderivative
of y squared over 2 minus y to the third over 3? Well the antiderivative of y
squared-- and you have to divide by 3, so it's
y cubed over 6. Minus y to the fourth--
you have to divide by 4. Minus y to the fourth over--
did I mess up some place? No, I think this is correct.
y to the fourth over 12. Oh wait. How did I get a 3 here? That's where I messed up. This is a 2, right? Let's see. x is square root of y. Yeah, this is a 2. I don't know how I ended up. Right? Square root of y squared
is y, times y squared y to the third over 2. Right. And then when I take
the integral of this it's 4 times 2. 8. Got to make sure I don't make
those careless mistakes. That's the tough part. I just want to make sure
that you got that too. I hate it when I do that. But I don't want to
re-record the whole video. So when I evaluated
this-- right. This is right, and then I take
the antiderivative of y to the third over 2, I get y
to the fourth over 8. And now I evaluate
this at 1 and 0. And that give us what? 1/6 minus 1/8 minus-- well both
of these when you evaluate them, are going to be 0's. So this is going to be
another 0 minus 0. So you don't have to
worry about that. So what's 1/6 minus 1/8? Let's see. 24. That's 4 minus 3 over 24, which
is equal to 1/24-- is the volume of our figure. So this time, the way we just
did it, we took the integral with respect to x first, and
then we did it with respect to y. Let's do it the
other way around. So let me erase some things. And hopefully I won't make
these careless mistakes again. I'll keep this figure, but
I'll even erase this one. Let me erase all of this stuff. We have room to work with. So I kept that figure. But let me redraw just the
xy plane, just so we get the visualization right. It's more important to
visualize the xy plane in these problems than it
is to visualize the whole thing in 3-d. That's the y-axis,
that's the x-axis. OK. Our upper bound, you can
say, is the graph y is equal to x squared. Or, you could view it as
the bound x is equal to the square root of y. That is x is equal to 1,
that's y is equal to 1. And we care about the volume
above the shaded region. Right? That shaded region is this
yellow region right here. And let's draw our da. I'll draw a little
square actually. And I'll do it in magenta. So that's our little da. And the height is dy. That's a y. dy. And it's with this dx. Right? So the volume above this little
square-- that's the same thing as this little square. Just like we said before, the
volume above it is equal to the value of the function. Right? The height is the value of the
function, which is x y squared. And then we multiply it
times the area of the base. Well the area of the base, you
could say it's da, but we know it's really dy times dx. I didn't have to write
that times there. You can ignore it. And I wrote the y first
just because we're going to integrate with
respect to y first. We're going to sum
in the y direction. So what does summing in
the y direction mean? It means we're going to
add that square to that square to that square-- [COUGHING] Excuse me. So we're adding all the
dy's together, right? So my question to you is, what
is the upper bound on the y? Well once again, we
bump into the curve. Right? So the curve is the upper
bound when we go upwards. And what is the upper
bound on the curve? Well we're holding x
fixed, so for any given x what is this point? Well that's going
to be x squared. Right? Because this is the graph
of y equals x squared. So our upper bound is y
is equal to x squared. And what's our lower bound? We can keep adding these
squares down here. We're adding all the
little changes in y. So what's our lower bound? Well our lower bound is just 0. That was pretty
straightforward. So this expression, as it is
written right now, is the volume above this rectangle. Right? Let me draw it. Is the volume above
this rectangle. These are the same rectangles. The volume above
that rectangle. Now what we want to do is add
up all the dx's together, and we'll get the volume above
the entire surface. So that rectangle, now
we'll add it to another dx one, dx one like that. So what's the upper and
lower bounds on the x's? We're going from x is
equal to 0, right? We don't bump into the graph
when we go all the way down. So we go from x is equal to 0. And then our upper bound
is x is equal to 1. So x is equal to 0,
x is equal to 1. And in general, one way to
think about it is when you're doing kind of the last sum or
the last integral, you really shouldn't have variable
boundaries at this point. Right? Because our final answer has
to be a number, assuming that we're not dealing with
something very, very abstract. But our final answer is
going to have a number. So if you had some variables
here you probably did something wrong. And it's always useful, I
think, to draw that little da and then figure out-- OK,
I'm summing to dy first. When I go upwards I
bump into the curve. What is that upper bound
if x is constant? Oh, it's x squared. y
is equal to x squared. If I go down I bump into the
x-axis, or y is equal to 0. And so forth and so on. So now let's just evaluate
this and confirm that we get the same answer. So we're integrating with
respect to y first. So that's x y to the
third, over three. At x squared and 0. And then we have our outer
integral. x goes from 0 to 1. dx. If we substitute x
squared in for y. x squared to the third
power is x to the sixth. x to the sixth times
x-- let me write that. So we have x times x squared
to the third power, over 3. That equals x to the seventh. Right? x squared to the third--
you multiply the exponents, and then you add these. x to the
seventh over 3, minus this evaluated with y is 0. But that's just going to be 0. Right? And then we evaluate
that from 0 to 1, dx. We're almost there. Increment the exponent. You get x to the eighth over 8. And we already have a 3 down
there, so it's over 24. And you evaluate
that from 0 to 1. And I think we get
the same answer. When you evaluate it at
1 you get 1/24 minus 0. So once again, when we
integrate in the other order you still get the volume of
the figure, being 1/4, whatever, cubic units. Anyway, see you in
the next video.