   Chapter 5.3, Problem 1E

Chapter
Section
Textbook Problem

# Let S be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume V of S. Sketch a typical approximating shell. What are its circumference and height? Use shells to find V. To determine

To find:

The volume of the solid obtained by rotating the region shown in the figure about the y-axis  and sketch the typical approximating shell.

Explanation

1) Concept:

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

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

V= ab2πxf(x)dx

Where 0ab

2) Given:

The region bounded by y=xx-12 rotated about the y- axis.

3) Calculation:

As the region is bounded by y=xx-12 rotated about the y-axis,

let us consider the problem of finding the volume by the washer method.

If we slice perpendicular to y-axis then we get a washer.

To compute the inner radius and the outer radius of the washer, we would have to solve the cubic equation.

y=xx-12 for x  in terms y that’s not easy.

that is, to find the functions g1and g2 as shown in fig is different.

Using the shell method, the typical approximating shell with radius x is shown below.

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