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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.

x = 2 y 2 ,   y 0 ,   x = 2 ;    about y = 2

To determine

To find:

The volume generated by rotating the region bounded by the given curves about y=2 using the cylindrical shells.

Explanation

1) Concept:

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

ii. By the shell method, the volume of the solid by rotating the region under the curve x=f(y) from a to b, about x -axis is

V= ab2πyf(y)dy

where,  0a<b

2) Given:

x=2y2, y0, x=2; about y=2

3) Calculation:

As the region is bounded by x=2y2, y0, x=2, and rotated about y=2, draw the region using the given curves.

The graph shows the region and the height of typical cylindrical shell formed by the rotation about the line y=2

It has the radius =2-y

Therefore, the circumference is 2π(2-y) and the height is x=2-2y2

To find the points of intersection, equate x=2 and x=2y2. Thus

2y2=2

y=-1, 1

y=-1 is not in the domain

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