Theory and ExamplesEquivalence of the washer and shell
Chapter 6, Problem 45E(choose chapter or problem)
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)
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