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## Volume: shell method (optional)

# Calculating integral with shell method

## Video transcript

In the last video we
were able to set up this definite integral using
the shell or the hollow cylinder method in order to
figure out the volume of this solid of revolution. And so now let's just
evaluate this thing. And really the main
thing we have to do here is just to multiply
what we have here out. So multiply this expression out. So this is going to be equal
to-- I'll take the 2 pi out of the integral. 2 pi times the
integral from 0 to 1. Let's see, 2 times
the square root of x is 2-- I'll write it
as 2 square roots of x. But I'll write it
as 2x to the 1/2. It'll make it a
little bit easier to take the antiderivative
conceptually, or at least in our brain. So two times the square
root of x is 2x to the 1/2. 2 times negative x squared
is negative 2 x squared. And then we have negative x
times the square root of x. Well, that's x to the
first times x to the 1/2. That's going to be negative
x to the 3/2 power. And then we have negative
x times negative x squared that's going to be
positive x to the third power. And all of that dx. And so now we're ready to
take the antiderivative. So this is going to be equal to
2 pi times the antiderivative of all of this business
evaluated at 1 and at 0. So the antiderivative
of 2 times x to the 1/2 is going to be 2-- it's
going to be-- let's see. We're going to take x
to the 3/2 times 2/3. So it's going to be
4/3 x to the 3/2. And then for this
term right over here it's going to be negative
2/3 x to the third. And you could take
the derivative here to verify that you
actually do get this. And then right over here, let's
see, if we incremented this, you get x to the 5/2. And so we're going to
want to multiply by 2/5. So minus-- let me do
this in another color. Let's see, so this one
right over here, it's going to be minus 2/5
x to the 5/2 power. Yep, that works out. And then finally you're going
to have x to the fourth over 4 plus-- let me do that
in a different color-- plus x to the fourth over 4. That's this term
right over here. And now we just have
to evaluate at 1 and 0. And 0, luckily, all of these
terms end up being a 0. So that's nice and cancels out. And so we are just
left with-- we're just-- [INAUDIBLE] cancel out. It just evaluates to 0. So this is just 2 pi times when
you evaluate all this business at 1. So that's going to be 4/3
minus 2/3 minus 2/5 plus 1/4. And the least common
multiple right over here looks like 60, so
we're going to want to put all this over
a denominator of 60. So it's going to be 2 pi
times all of this business over a denominator of 60. And 4/3 is same thing as 80/60. Negative 2/3 is the same
thing as negative 40/60. Negative 2/5 is the same
thing as negative 24/60. And then 1/4 is the
same thing as 15/60. So this is equal to--
and actually this will cancel over
here, and you'll just get a 30 in your denominator. So in your denominator,
you get a 30. And up here 80 minus 40 is 40. 40 minus 24 gets us to 16. 16 plus 15 is 31. So we get 31 times pi
over 30 for the volume of the figure right over there.