   Chapter 7.1, Problem 63E

Chapter
Section
Textbook Problem

# Use the method of cylindrical shells to find the volume generated by rotation the region bounded by the curves about the given axis. y = e − x ,   y = 0 ,   x = − 1 ,   x = 0 ; about x = 1

To determine

To evaluate: the volume generated by rotating the region bounded by the given curves about the given axis using the cylindrical shell method.

Explanation

Consider a function y=f(x) between the points x=a and x=b. If the region under the curve y=f(x) between a and b is rotated about the y-axis, a solid shape is obtained. The volume of that solid shape can be calculated by taking the volume of small cylindrical shells making up the solid and then adding them together.

Formula used:

Volume of the solid obtained by rotating the region from a to b under the curve y=f(x) is given by the following integral:

V=ab2πxf(x)dx,           where 0a<b

Given:

The curve,y=ex

Bounded by y=0,x=0,x=1

Calculation:

Since the axis of rotation is about x=1, and it is bounded by x=0, the volume required will be the difference in volume of solid generated by rotating y=ex from -1 to 1 and the solid generated from rotating y=ex from 0 to 1.

V=102π(1x)exdx=2π(10exdx10xexdx) …… (1)

Solve the integral 10xexdx using integration by parts. Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate

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