   Chapter 7.1, Problem 62E

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 = e − x ,   x = 1 ; about the y-axis

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 curves,y=ex,y=ex

Bounded by x=1

Calculation:

The graph for the two curves along with the line x=1 is shown below:

The triangle like region is the region bounded by the two curves and the line, which is being rotated. So here, the range for x will be from 0 to 1 and the height of the cylinder will be exex.

Substitute the curve into the volume formula with a and b as 0 and 1 respectively;

V=012πx(exex)dx=2π(01xexdx01xexdx) …… (1)

Solve the integral 01xexdx using integration by parts. Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate. Let

u=x      dv=exdx

Then, the differentiation of u and antiderivative of dv will be

du=dx     v=ex

The formula for integration by parts for definite integral is given by

abf(x)g(x)dx=f(x)g(x)]ababg(x)f(x)dx

So, the integration will become:

01xex</

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