   Chapter 5.3, Problem 12E

Chapter
Section
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 = − 3 y 2 + 12 y − 9 ,    x = 0

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 solid by rotating the region under the curve y=f(x) about x-axis from a to b is

V= ab2πyf(y)dy

where,  0ab

2) Given:

The region bounded by x=-3y2+12y-9, x=0 rotated about the x- axis.

3) Calculation:

As the region is bounded by x=-3y2+12y-9, x=0 using shell method, find the typical approximating shell with the radius  y.

Therefore, the circumference is 2πy and the height is x=-3y2+12y-9

a=1 and b=3

So, the total volume is

V= ab2πy [-3y2+12y-9]dy

V= 132π [-3y3+12y2-9y ] dy

V=2π13-3

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