We have to evaluate the following triple integral in the region $y = 2x^2+2z^2$ to $y=8$
$$\iiint \sqrt{3}\sqrt{x^2+z^2} dV$$
I used cylindrical coordinate $x = \rho \cos \phi$, $z = \rho \sin \phi$, $y=y$ to get:
$$\int_{\rho = 0}^{2}\int_{\phi = 0}^{2\pi}\int_{y = 2\rho^2}^{8} \sqrt 3 \rho^2\, dy\, d\phi\, d\rho$$
But this gives answer as: $\frac{256 \pi \sqrt 3}{15}$ but answer is clearly given to be $2\pi/3$.
Is there some resource like mathematica/wolfram alpha which I can use to check answer to such exercises
Thank you!