   Chapter 5.3, Problem 42E

Chapter
Section
Textbook Problem

# The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x = ( y − 3 ) 2 ,   x = 4 ;    about   y = 1

To determine

To find:

The volume of the solid generated by rotating the region bounded by the given curves about specific axis by any method.

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 line y=l from a to b is

V= ab2π(y-l)g(y)dy

where 0ab and g(y)=x

2) Given:

The region bounded by x=y-32 , x=4 and rotated about the line y=1

3) Calculation:

As region is bounded by x=y-32, x=4 and rotated about the line y=1

Using the shell method,

Radius is (y-1) the circumference is 2π(y-1) and the height is 4-y-32

From the graph, y=1 and y=5 are the y coordinates of the intersection of these two curves.

Therefore, a=1 and b=5

So, the total volume is

V= ab2π(y-1)[4-y-32]dy

V=2π

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