   Chapter 5.3, Problem 24E

Chapter
Section
Textbook Problem

# (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.(b) Use your calculator to evaluate the integral correct to five decimal places. y = x ,   y = 2 x / ( 1 + x 3 ) ;    about  x = − 1

To determine

a)

To find:

The integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.

Explanation

1) Concept:

i. If x  is the radius of the typical shell, then the circumference =2πx and the height is y=f(x)

ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about x-axis from a to b is

V= ab2πx f(x)dx

where  0a<b

2) Given:

3) Calculation:

As the region is bounded by

Draw the region using the given curves.

The graph shows the region. A typical cylindrical shell is formed by the rotation about the line  x=-1.

Therefore, the circumference is

2πx+1 and the height is 2x1+x3-x

Now, to find the intersection points,

2x1+x3=x</

To determine

b)

To evaluate:

The integral ofthe volume of the solid.

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