Richard P. answered 03/10/15
Fairfax County Tutor for HS Math and Science
The disk method looks to be easier than the washer method for this problem.
The integral over x is from -2 to 2 over a stack of thin disks each with volume π (4 - x2)2 dx
This integral can be carried out by expanding the squared binomial and then integrating term by term.
I got ( 512/15) π