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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 y-axis.

y = x 2 ,    y = 6 x 2 x 2

To determine

To find:

The volume generated by rotating the region bounded by given curves about y-axis using cylindrical shells

Explanation

1) Concept:

i. If x is radius of typical shell then circumference =2πx and height is y

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

V= ab2πxf(x)dx

Where,  0ab

2) Given:

The region bounded by y=6x-2x2, y=x2 rotated about the y- axis.

3) Calculation:

As the region is bounded by y=6x-2x2, y=x2 rotated about the y- axis,

Using the shell method, the typical approximating shell with the radius x is shown below,

Therefore, the circumference is 2πx and the height is 6x-2x2-x2

 that is, -3x2+6x

To find the values of a and b, consider 6x-2x2=x2

6x-2x2-x2=0

-3x2+6x=0

That is, 3x2-x=0

x=

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