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PROFESSOR: Welcome
back to recitation.
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In this video, what
we'd like to do
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is find the volume
of a paraboloid--
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this one that I've drawn
on the board-- using
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what we know about Riemann
sums and integrals.
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And so, this paraboloid,
just so you understand,
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what we do is we take the
curve y equals x squared
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and we rotate it
around the y-axis.
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And we're looking in
particular, from height--
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the height of capital H,
so from 0 to capital H,
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that's what our y-value
is ranging over.
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And what we'd like to do
is find the volume enclosed
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by 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
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about how you could write
the volume as an integral.
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And I would suggest
that you think
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about integrating in the
y variable as opposed
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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,
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and then 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
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is find the volume
of this 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
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we start the process.
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And what I want to do is break
up this paraboloid first,
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into 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,
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so 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
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of right here coming around
that way, going down,
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so it sticks out a
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-- well,
that doesn't look very even.
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But we'll just say
that there are four.
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To keep it with four,
I'll say the last one
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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
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and 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
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the volume of the cylinders.
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And then what we'd like to do
is to get better approximations,
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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,
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we're going to get better
and better estimates
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of the paraboloid's volume.
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And then in the
limit, we will get
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the volume of the paraboloid.
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So this again should
remind you, this
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sounds a lot like Riemann sums.
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What we've been doing
with Riemann sums.
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You're adding up--
you're estimating area
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under the curve, in
the case that you've
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seen mostly in class.
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But maybe this reminds
you of the volume
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of the pyramid 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 four
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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
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of the 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 it,
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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 bigger
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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 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 y,
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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 cylinder
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at height y-- I'm using the
upper value here-- so at height
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y, what is 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
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is not the height
on the whole thing;
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this is the height of
that 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
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by taking the y-value
here, multiplying it by pi
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and then multiplying
it by delta y.
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And I do that for this one, this
one, this 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 paraboloid.
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So now what can I do?
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Well, again, I mentioned we can
divide into more subintervals.
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And let's say we
only divide-- let's
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say we divide it into
n 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
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the volume of the whole thing.
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I'm going to take a sum
from i equals 1 to n
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of 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 height
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where I'm taking my cylinder.
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Now delta y, in this
case, if I divide
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into n equal subintervals,
delta y is H divided by n.
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And then you can actually figure
out the y 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 one
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would be H*n over n and you'd
have 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,
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this 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
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sub i 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
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an integral from
0 to H of pi*y*dy.
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And that is the
integral representation
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of the volume of the paraboloid.
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Now this is a very easy
thing to take an integral on,
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so you could actually
evaluate this and find it.
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I believe you'll get something
like pi y squared over 2 from 0
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to H. So you should
get something
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like pi over 2
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
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x squared, from, or
at height capital H.
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So that's where we'll stop.