WEBVTT
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So for this problem, we're going to be using
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the washer method. We have to integrate along the
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Y axis, focusing on the inner diameter and the
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outer diameter. So we're going to use symmetry and
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double the integral. So rather than doing it from
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a negative square root to a positive square root,
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we'll just go from y equals zero to y equals
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the square root of R squared minus R squared.
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Bigger and little are and then the upper limit is
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going to be found by using the Pythagorean theorem on
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the right triangle in the following picture. So what
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we have is this we have our circle. Um
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, there's this smaller radius right here. This smaller
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radius is the little r. And then the larger
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radius right here is the big are we also have
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a larger radius straight here, Obviously, which is
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going to be the same since its circle. Um
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, so we want to solve the circle equation for
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X. So first, what we have is r
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squared equals X squared plus y squared. So solving
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for X, we're going to get the X equals
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the square of R squared minus y squared. Then
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that means that our outer radius will be are one
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and that is R squared minus y squared and then
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our inner radius we already see as just are.
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So now we're going to integrate with the washer formula
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That's the equals pi times the bounds Where'd you decided
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that the bounds are zero to the square root of
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R squared minus y squared And then we plug in
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our our one squared minus r two squared. Once
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we do that, we end up being able to
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simplify a little bit further and it's going to be
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r squared minus y squared minus little r squared.
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Do you mind simplifying this further? Using integral zoned
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properties, we know we get four thirds our big
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R squared minus little r squared to the three halves
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. Um, and then we have the cylindrical shell
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method that we could also use. So what we
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have is r squared equals X squared plus y squared
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. And in this case, what we have is
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solving for why we'd get the square root of r
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squared my x squared, Um, and this is
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using this special method, but it would end up
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giving us the same answer and that is a volume
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of four thirds pie, um, four thirds pi
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. And that should be here to four thirds pi
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four thirds pi r squared minus little r squared to
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the three halves.