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Math
CalculusCalculus (MindTap Course List)8th Edition

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Chapter 5.3, Problem 43E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
Problem 43E
Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

x = ( y − 1 ) 2 ,   x − y = 1 ;    about  x = − 1

To determine

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

Explanation

1) Concept:

i. If the cross section is a washer with inner radius rin and outer radius rout, then 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 solid revolution about y-axis,

V= ∫abA(y)dy

2) Given:

The region bounded by x=y-12, x-y=1  rotated about the x=-1

3) Calculation:

Find the volume by using washers.

The graph of the region bounded by the given curves is

From figure, as region rotate about the line x=-1, so strip is perpendicular to y-axis.

A cross section of the solid is washer with outer radius is 1+y--1= y+2  and inner radius is y-12-(-1)

From the graph, the points of intersection are (1, 0) and (4, 3)

So its area is given by,

Ay=πouter radius2-πinner radius2

Ay=2+y2-(y-12-(-1)) 2

V= ∫03Aydy=∫03[π2+y

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Chapter 5.3,
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Sect-5.1 P-1ESect-5.1 P-2ESect-5.1 P-3ESect-5.1 P-4ESect-5.1 P-5ESect-5.1 P-6ESect-5.1 P-7ESect-5.1 P-8ESect-5.1 P-9ESect-5.1 P-10E
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