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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

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

y = 4 x x 2 ,   y = 3 ;    about  x = 1

To determine

To find:

The volume generated by rotating the region bounded by the given curves about x=1 using the cylindrical shells.

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=4x-x2, y=3, rotate about x=1

3) Calculation:

As the region is bounded by y=4x-x2, y=3, and rotated about x=1, draw the region using the given curves.

The graph shows the region and the vertical cross section of cylindrical shell formed by the rotation about the line x=1.

It has the radius =x-1

Therefore, the circumference is 2π(x-1) and height is (4x-x2)-3

To find the point of intersection, equate both curves

4x-x2=3

x2-4x+3=0

This is a quadratic equation.

Solve by using the quadratic formula

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