   Chapter 5.2, Problem 3E

Chapter
Section
Textbook Problem

# Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x − 1 ,   y = 0 ,   x = 5 ; about the x-axis

To determine

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the x axis and sketch the region, the solid, and a typical disk or washer.

Explanation

1) Concept:

i. If the cross section is a disc and the radius of the disc is in terms of x  or y then the area A=π radius2

ii. The volume of the solid revolution about the x-axis is

V= abA(x)dx

2) Given:

The region is bounded by y =x-1,  y=0, x=5; about the x- axis.

3) Calculation:

The region is bounded by y=x-1,  y=0, x=5  and solid is obtained by rotating it about the x- axis. It is shown below:

Here, the region rotated about the x – axis, so the cross-section is perpendicular to the x-axis.

A cross section of the solid is a disk with radius x-1.

So, its cross sectional area becomes

Ax=πx-12=πx-1

The solid lies between  x=1 and x= 5

Therefore, the volume of the solid is

V= 15A

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