Find volume of the ellipsoid $\displaystyle x^2 +2(y^2+z^2) \le 10$ intersected by the cylinder $\displaystyle y^2 = z^2 \le 1 $

seems like changing to cylindrical coordinates would be best

so I have

$\displaystyle \left\{ \begin{array}{cc} r^2cos\theta + 2r^2sin^2\theta +2z^2 \le 10 \\r^2sin^2\theta +z^2 \le 1 \end{array}\right.$

$\displaystyle 0 \le \theta \le 2\pi$

and solving for z $\displaystyle -\sqrt{1-r^2sin^2\theta} \le z \le \sqrt{1-r^2sin^2\theta}$

and $\displaystyle -\sqrt{8}sec\theta \le r \le \sqrt{8}sec\theta$

before I set up the integrals, are those the correct bounds?

edit: I tried on wolfram|alpha but it seems to not work