Volume of Revolution for Curve y=(12/(x + 3))- 4 Rotated about the y-axis

[jack1234]

The question is

Given a curve C whose equation is give by y=(12/(x + 3))- 4,
find the region bounded by C, the line x=0, the line x=3 and the line y=8 is rotated through 360 degree about the y-axis. Calculate the exact value of the volume generated.

My solution is here
http://www.geocities.com/myjunkmail31/Volume.jpg

However, this solution is not the answer in the textbook. The anwser in the textbook is
36pi(1+2ln2).
Can someone show me where am I doing wrong?

 

[benorin]

The error is in V₁: Easy fix: $$V_1 = \pi 3^2 (2) - \pi (54 - 2 ln 2)$$.

The volume you calculated is that of the solid generated by rotating the region between the given curve and the y-axis about the y-axis, but you wanted to rotate the region between the given curve and x=3 about the y-axis. So just subtract what you got for V₁ from the volume of the cylinder of radius 3 with height 2.

 

[jack1234]

Awesome, very thanks=)