# Simple shell

1. Feb 21, 2005

### pattiecake

"simple" shell

I know this is relatively simple, but I'm a little rusty. Could someone help me out? We want to find the volume of the solid obtained by rotating the region bounded by the curves y=x^4 and y=1 about the line y=7 using the cylindrical shell method.

According to my book the general formula for cylindrical shell method is: V=(circumference)(height)(thickness) or (2pi*r)(r*h)(delta r). So I set up the integral as (2pi) integral [7x -x^5] dx. The boundaries are found by setting x^4=1, which yields -1 and 1. After differentiating we have 2pi[7/2x^2-1/6x^6] from -1 to 1. Because of my boundaries, I initially got the volume=0, but I don't think that's possible. I assumed the minus sign should be a plus, but after adding and multiplying by 2pi, I still got the wrong answer.

Any clues would be much appreciated! Thanks!

Last edited: Feb 21, 2005
2. Feb 21, 2005

### Galileo

You're rotating the region about the line y=7 right?
Then if you're using cylindrical shells, you should integrate wtr y (from 0 to 1).
The radius of the shell is $7-y^{1/4}$.

Try setting up the integral again.