# Integration to Form a Solid - Using Shells?

1. Jan 25, 2009

### x^2

Integration to Form a Solid -- Using Shells?

1. The problem statement, all variables and given/known data

Use the shell method to find the volumes of solids generated by revolving the shaded regions
about the indicated axes. The graph shows the equations:

$$x = \dfrac{y^4}{4}-\dfrac{y^2}{2}$$
and
$$x = \dfrac{y^2}{2}$$

a) The x-axis
b) The line y = 2
c) The line y = 5
d) The line y = -5/8

2. Relevant equations

Shell method: $$V = 2\pi\int_a^b \! y * f(y) \, dy$$
3. The attempt at a solution
I attempted to use the shell method to find the volume of the formed "bowl" but I get a negative number:

$$V = 2\pi\int_0^2 \! y * (\dfrac{y^4}{4}-\dfrac{y^2}{2} - \dfrac{y^2}{2}) \, dy = 2\pi\int_a^b \! y * (\dfrac{y^4}{4}-y^2) \, dy = 2\pi\int_a^b \! (\dfrac{y^5}{4}-y^3) \, dy = 2\pi[\dfrac{y^6}{24} - \dfrac{y^4}{4}]^2_0 = 2\pi[\dfrac{64}{24} - \dfrac{16}{4}]$$

Where am I going wrong?

Thanks,
- x^2