   Chapter 5.3, Problem 9E

Chapter
Section
Textbook Problem

# Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. x y = 1 ,    x = 0 ,    y = 1 ,    y = 3

To determine

To find:

The volume generated by rotating the region bounded by the given curves about the x-axis using the cylindrical shells.

Explanation

1) Concept:

i. If y  is the radius of the typical shell then the circumference =2πy and the height is 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= ab2πyf(y)dy

where,  0ab

2) Given:

The region bounded by xy=1, x=0, y=1, y=3 rotated about the x- axis.

3) Calculation:

As the region is the bounded by xy=1, x=0, y=1, y=3 rotated about the x- axis,

xy=1x=1y

Using the shell method, the typical approximating shell with the radius y is

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