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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 of the solid obtained by rotating the region bounded by the given curves about the x-axis.

x + y = 4 ,    x = y 2 4 y + 4

To determine

To find:

The volume generated by rotating the region bounded by the given curves about x-axis 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) about x- axis from a to b is

V= ab2πy f(y)dy

where,  0ab

2) Given:

The region bounded by x+y=4 , x=y2-4y+4   rotated about the x- axis.

3) Calculation:

As the region is bounded by x+y=4 x=4-y, x=y2-4y+4 

Using the shell method, the typical approximating shell with the radius y is

To find the height of the strip, subtract the above functions.

4-y-(y2-4y+4)

=-y2+3y

Therefore, the circumference is 2πy and the height is (-y2+3y)

To find the point of intersections,

4-y=(y2-4y+4)

4-y-(y2-4y+4)=0

-y2+3y=0 

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