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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 = x 3 ,   y = 8 ,   x = 0 ;    about  x = 3

To determine

To find:

The volume generated by rotating the region bounded by the given curves about x=3 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.

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,  axb

2) Given:

y=x3, y=8, x=0, rotate about x =3

3) Calculation:

As the region is bounded by  y=x3, y=8, x=0, and rotated about  x=3, draw the region using the given curves.

The graph shows the region and the cylindrical shell formed by rotation about the line x=3

It has the radius y=3-x

Using the shell method,

Find the typical approximating shell with radius  3-x,

Therefore, circumference is 2π(3-x) and height is 8-x3

Find the points of intersection (the limit of integrals). At intersection

x3=8

So

x=2

So, a=0 and b=2

So, the volume of the given solid is

V= ab2π 3-xf<

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