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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

(a) Suppose S is a solid with known cross-sectional areas. Explain how to approximate the volume of S by a Riemann sum. Then write an expression for the exact volume.

(b) If S is a solid of revolution, how do you find the cross-sectional areas?

(a)

To determine

To explain:

How to approximate the volume of solid S by Riemann sum and write an expression for exact volume.

Explanation

Slice the given solid S into n“slabs” of equal width x.

If xi* is sample point on ith slab and Axi* is the cross-sectional area of S at xi*, the volume of the ith slab is approximately Axi*x.

Therefore, the volume of S is approximately

i=1nA(xi*)x

For an exact volume of solid slab S that lies between x=a and x=b

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