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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

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.

(b) Use your calculator to evaluate the integral correct to five decimal places.

y = sin x ,   y = 0 ,   x = 2 π ,   x = 3 π ;    about the  y -axis

To determine

a)

To find:

The integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.

Explanation

1) Concept:

i. If x  is the radius of the typical shell, then the circumference =2πx and the height is y=f(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= ab2πx f(x)dx

Where,  0a<b

2) Given:

y=sinx, y=0, x=2π, x=3π, rotate about y-axis

3) Calculation:

As the region is bounded by y=sinx, y=0, x=2π, x=3π, and rotated about y-axis, draw the region using the given curves

To determine

b)

To evaluate:

The integral ofthe volume of the solid

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