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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
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 2 = x ,   x = 2 y ; 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 specified line and sketch the region, the solid, and a typical disk or washer.

Explanation

1) Concept:

i. If the cross section is a washer with the inner radius rin and the outer radius rout, then the area of washer is obtained by subtracting the area of the inner disk from the area of the outer disk.

A=π outer radius2 -π inner radius2

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

V= abA(y)dy

2) Given:

y2=x,  x=2y ; about the y- axis

3) Calculation:

The region bounded by y2=x, x=2y and the solid obtained by rotation about the y- axis is shown below:

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

A cross section of the solid is the washer with the outer radius 2y and the inner radius y2.

So, its cross sectional area is

Ay=πouter radius2-πinner radius2 =π2y2-πy22

Ay=π4y2-y4

The region of integration is bounded by y2=x and x=2y

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