Theory and ExamplesEquivalence of the washer and shell

Chapter 6, Problem 45E

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Problem   45E

Theory and Examples

Equivalence of the washer and shell methods for finding volume Let ƒ be differentiable and increasing on the interval with a > 0, and suppose that ƒ has a differentiable inverse, f –1Revolve about the y-axis the region bounded by the graph of ƒ and the lines x = a and y = ƒ(b) to generate a solid. Then the values of the integrals given by the washer and shell methods for the volume have identical values:

To prove this equality, define

Then show that the functions W and S agree at a point of [a, b] and have identical derivatives on [a, b]. As you saw in Section 4.8, Exercise 128, this will guarantee W(t) = S(t) for all t in [a, b]. In particular, W(b) = S(b)