Using shell method to find the volume of a solid

[mmont012]

Homework Statement

Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y=2x+15 and the parabola y=x² about the following lines:

a) The line x=5 b) The line x= -3 c) The x-axis d) The line y=25

Note: leave answer in terms of pi

Homework Equations

v=2pi∫(shell radius)(shell height)

The Attempt at a Solution

I know that I am making a mistake somewhere; I have a feeling that it is in my set... I am hoping that someone will be able to point it out to me. I am stuck on part a) and if I figure out my mistake I am confident that I will be able to do the other parts of the problem.

First thing that I did (after graphing which is attached) was find the limits of integration:
2x+15=x²
x²-2x-15
(x-5)(x+3)
So the limits of integration are from -3 to 5.

v=2pi∫(5-x)(2x+15-x²)dx

v=2pi∫10x+75-5x²-2x²-15x+x³

v=2pi∫75-7x²-5x+x³

v=2pi(75x-(7/3)x³-(5/2)x²+(1/4)x⁴

(plug in the limits of integration)
v=2pi [(2125/12)+(607/4)]

v=2pi(1973/6)

v=1973pi/3 <---this answer is wrong.

I hope that someone will be able to help me with this, thanks for stopping by and sorry for the crappy paint graph.

 

[slider142]

v=2pi(75x-(7/3)x³-(5/2)x²+(1/4)x⁴

(plug in the limits of integration)
v=2pi [(2125/12)+(607/4)]

Your setup is fine, but your value for the primitive function at x=-3 is not correct. Try computing that particular value again. :)