WEBVTT

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PROFESSOR: Welcome back
to recitation.

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In this video, what we'd like to
do is find the volume of a

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paraboloid--

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this one that I've drawn
on the board--

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using what we know about Riemann
sums and integrals.

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And so, this paraboloid, just so
you understand, what we do

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is we take the curve y equals
x squared and we rotate it

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around the y-axis.

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And we're looking in particular,
from height, the

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height of capital H. So from 0
to capital H, that's what our

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y value is ranging over.

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And what we'd like to do is find
the volume enclosed by

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the rotation of y equals x
squared around the y-axis.

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So what I'd like you to do is
to think about how you could

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write the volume
as an integral.

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And I would suggest that you
think about integrating in the

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y variable as opposed
to the x variable.

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So we're going to do it
in the y variable.

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So I'll give you a little while
to think about that.

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Start as a Riemann sum, take
it to an integral, and then

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I'll come back and show
you how I did it.

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OK, welcome back.

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So again, what we're trying to
do is find the volume of this

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paraboloid.

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And we're hoping to write it
as an integral in the end.

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So I'm going to give us a little
understanding of how we

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start the process.

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And what I want to do is break
up this paraboloid first, into

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our first-best guess.

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And I'm going to estimate the
volume using four different--

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I'm breaking it up into
four pieces, so

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I'll use four cylinders.

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So let me actually just
draw these cylinders.

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So I'm going to start
at the top radius.

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And that would maybe be
my first cylinder.

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And then the next radius would
be the one down here.

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Actually maybe I can
alternate color.

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You'd be able to see them
a little better.

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The next radius would actually
be this one kind of right here

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coming around that way, going
down, so it sticks out a

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little bit from there.

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Let me just kind of shade
that lightly.

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And the next one is using
this cylinder.

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So that's our diameter here.

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And then the last one--

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well, that doesn't
look very even.

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But we'll just say that there
are four to keep it with four.

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I'll say the last one is
starting right here.

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So I don't have to draw
any more pictures.

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And the last one is
that cylinder.

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So there are four
cylinders here.

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This one comes around behind.

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There are four cylinders here.

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There's the blue one here, a
white one, a red one and

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another blue one.

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And I can actually estimate the
volume of this paraboloid

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by finding the volume
of these cylinders.

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So we have to understand
how to find the

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volume of the cylinders.

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And then what we'd like
to do is to get better

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approximations, we're going to
make these cylinders flatter.

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So we're going to make
them not as tall.

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And then as they get flatter and
flatter and flatter, we're

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going to get better and better
estimates of the

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paraboloid's volume.

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And then in the limit, we will
get the volume of the

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paraboloid.

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So this again should remind you,
this sounds a lot like

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Riemann sums. What we've been
doing with Riemann sums.

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You're adding up your estimating
area under the

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curve in the case that you've
seen mostly in class.

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But maybe this reminds you of
the volume of the pyramid

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problem that you saw on
the lecture videos.

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So it's very similar
to that idea.

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Now let's figure out how we can
write this sum of these

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four cylinders in terms of y.

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So what do we need to do?

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Well, first we're going to
designate the height of the

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cylinder as delta y.

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That's the change in y.

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That does not look like a y.

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Let me try that again.

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Delta y.

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OK.

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So each height-- and even though
it doesn't look like

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it, we're going to say that
they're evenly divided.

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So let's make, in your mind
this one should be a bit

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bigger and this one should
be a bit smaller.

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So delta y is going
to be a constant.

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And then I have to figure
out the radius.

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Right?

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Well, we know that at
height y the radius

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squared is height y.

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Right?

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r squared equals h.

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So what I actually need is that
at height h, or at height

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y, the radius is square root y.

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Right?

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So at height y radius is
the square root of y.

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So when I want to look
at the volume of the

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cylinder at height y--

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I'm using the upper value here--
so at height y, what is

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the volume of the cylinder?

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We know in general the volume
is equal to pi r squared h.

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And so, the height here of the
cylinder-- this is not the

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height on the whole thing; this
is the height of that

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individual cylinder--

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the height of the cylinder
is delta y.

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And r is square root y.

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So r squared is just y.

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So the volume of a single
cylinder is pi y delta y.

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So again, we find the volume of
this cylinder by taking the

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y value here, multiplying it by
pi and then multiplying it

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by delta y.

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And I do that for this
one, this one, this

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one, and this one.

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And then I add them up.

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And that's my estimate of
the volume of the whole

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paraboloid.

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So now what can I do?

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Well, again, I mentioned
we can divide into more

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subintervals.

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And let's say we only divide,
let's say we divide it into n

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subintervals. right?

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What would this look like?

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I'm going to be taking a sum
to find the volume of the

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whole thing.

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I'm going to take a sum from
i equals 1 to n of

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these kinds of things.

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Pi, y sub i delta y.

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So each y sub i represents the
y value at the different

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height where I'm taking
my cylinder.

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Now delta y, in this case,
if I divide into n equals

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subintervals, delta y
is H divided by n.

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And then you can actually
figure out the y

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sub i's from that.

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The first one would
be H over n.

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The second one would be 2H over
n and so on until the top

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one would be Hn over
n and you'd have

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H as the top height.

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So those are the values that
we're ranging over.

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And now, as we let n go
to infinity, this

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is our Riemann sum.

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This is actually going
to be an integral.

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Delta y goes to dy.

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And let's look at
what happens.

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So we get dy here.

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We're going to have
an integral.

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And what are we actually
evaluating?

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We're evaluating the function
pi y at what places?

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Well, as delta y goes to 0,
the first value of y sub i

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that I get is pushing
down to 0.

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And the last value we know is H.
So we're actually taking an

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integral from 0 to
H of pi y dy.

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And that is the integral
representation of the volume

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of the paraboloid.

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Now this is a very easy thing to
take an integral on, so you

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could actually evaluate
this and find it.

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I believe you'll get something
like pi y squared over 2 from

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0 to H. So you should get
something like pi over 2

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capital H squared.

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You can check my math, but
I think that's correct.

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So that is actually the volume
of the paraboloid carved out

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by this particular curve, y
equals x squared, from, or at

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height capital H. So that's
where we'll stop.