   Chapter 5.2, Problem 4E

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 = 25 − x 2 ,   y = 0 ,   x = 2 ,   x = 4 ; 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 area A=π radius2

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

iii.  V= abA(x)dx

2) Given:

y=25-x2,  y=0, x=2, x=4 about the x- axis.

3) Calculation:

The region is bounded by y=25-x2,  y=0, x=2, x=4 and rotated about the x- axis is shown in figure:

Here, the region is 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 25-x2.

So, its cross sectional area becomes

Ax=π25- x22=π25-x2

The solid lies between  x=2 and x= 4

Therefore, the volume of the solid revolution about x</

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