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CalculusCalculus10th Edition

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Chapter 8, Problem 22PS
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Volume Consider the shaded region between the graph of y = sin x , where 0 ≤ x ≤ π , and the line y = c , where 0 ≤ c ≤ 1 , as shown in the figure. A solid is formed by revolving the region about the line y = c .

(a) For what value of c does the solid have minimum volume?

(b) For what value of c does the solid have maximum volume?

Chapter 8, Problem 22PS, Volume Consider the shaded region between the graph of y=sinx, where 0x, and the line y=c, where

(a)

To determine

To calculate: The value of c that is formed by revolving the region of the solid about the line y=c that has minimum volume.

Explanation

Given:

For function y=sinx where x lies on interval [0,π] and the line y=c, where c lies on the interval [0,1]; the graph is as below. A solid is formed by revolving the region about the line y=c.

Formula used:

The integration property:

∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx

The differentiation formula:

ddx(xn)=nxn−1

Calculation:

Here, y=sinx.∴, x=sin−1y.

Also, given y=c. ∴, x=sin−1c.

By integrating the circle with radius c−sinx half the volume can be found.

∴, the volume is:

12V=∫0arcsin(c)π(c−sinx)2dx+∫arcsin(c)π2π(sinx−c)2dx=∫0arcsin(c)π(c2+sin2x−2csinx)dx+∫arcsin(c)

(b)

To determine

To calculate: The value of c that forms a solid by revolving the region about the line y=c has maximum volume.

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P-15RECh-8 P-16RECh-8 P-17RECh-8 P-18RECh-8 P-19RECh-8 P-20RECh-8 P-21RECh-8 P-22RECh-8 P-23RECh-8 P-24RECh-8 P-25RECh-8 P-26RECh-8 P-27RECh-8 P-28RECh-8 P-29RECh-8 P-30RECh-8 P-31RECh-8 P-32RECh-8 P-33RECh-8 P-34RECh-8 P-35RECh-8 P-36RECh-8 P-37RECh-8 P-38RECh-8 P-39RECh-8 P-40RECh-8 P-41RECh-8 P-42RECh-8 P-43RECh-8 P-44RECh-8 P-45RECh-8 P-46RECh-8 P-47RECh-8 P-48RECh-8 P-49RECh-8 P-50RECh-8 P-51RECh-8 P-52RECh-8 P-53RECh-8 P-54RECh-8 P-55RECh-8 P-56RECh-8 P-57RECh-8 P-58RECh-8 P-59RECh-8 P-60RECh-8 P-61RECh-8 P-62RECh-8 P-63RECh-8 P-64RECh-8 P-65RECh-8 P-66RECh-8 P-67RECh-8 P-68RECh-8 P-69RECh-8 P-70RECh-8 P-71RECh-8 P-72RECh-8 P-73RECh-8 P-74RECh-8 P-75RECh-8 P-76RECh-8 P-77RECh-8 P-78RECh-8 P-79RECh-8 P-80RECh-8 P-81RECh-8 P-82RECh-8 P-83RECh-8 P-84RECh-8 P-85RECh-8 P-86RECh-8 P-87RECh-8 P-88RECh-8 P-89RECh-8 P-90RECh-8 P-91RECh-8 P-1PSCh-8 P-2PSCh-8 P-3PSCh-8 P-4PSCh-8 P-5PSCh-8 P-6PSCh-8 P-7PSCh-8 P-8PSCh-8 P-9PSCh-8 P-10PSCh-8 P-11PSCh-8 P-12PSCh-8 P-13PSCh-8 P-14PSCh-8 P-15PSCh-8 P-16PSCh-8 P-17PSCh-8 P-18PSCh-8 P-19PSCh-8 P-20PSCh-8 P-21PSCh-8 P-22PS

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