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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 ,   x = y 2 + 1 ;    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 x=f(y)

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

V= ab2πxf(y)dy

where  0a<b

iii. The theorem of symmetric function:

Suppose f  is continuous on -a, a,

(a) If f  is a even function, then

-aafxdx=20afxdx

(b) If f  is an odd function, then

-aaf(x)dx=0

2) Given:

x=2y2, x=y2+1; about y=-2

3) Calculation:

As the region is bounded by x=2y2, x=y2+1; about y=-2, draw the graph of 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

Using the shell method, to find the typical approximating shell with the radius 2+y

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