Triple Integral

[parisse]

You can rewrite the initial integral using parity in z and get the right answer:

Code:

2*int(int(int(sqrt(x^2+z^2), z = 0 .. sqrt(y - x^2)),y,x^2,4),x,-2,2)

If you add parity in x, you will have to simplify the answer.

Code:

4*int(int(int(sqrt(x^2+z^2), z = 0 .. sqrt(y - x^2)),y,x^2,4),x,0,2)

Multiple definite integrals are (too?) hard to integrate symbolically because the inner integrals have parameters and that prevent checking antiderivative discontinuities checks.