   Chapter 5.2, Problem 5E

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. x = 2 y ,   x = 0 ,   y = 9 ; about the y-axis

To determine

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the y  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 y-axis is

V= abA(y)dy

2) Given:

x=2y,  x=0, y=9; about the y- axis

3) Calculation:

The region is bounded by x=2y,  x=0, y=9 and rotated about the y- axis. The solid thus obtained is shown below:

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

A cross section of the solid is a disk with radius  2y.

So, its cross sectional area becomes

Ay=π2y2=4πy

The solid lies between  y=0 <

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