I would use the divergence theorem.

Here's a similar exercise : http://www.mathhelpforum.com/math-he...e-theorem.html, actually is the flux passing by .

So in your exercise I'd first calculate the divergence of the field and then chose an appropriate change of variables in order to evaluate .

Here's another thread where the volume of an elliptical cone is calculated. That might really help you here.

http://www.mathhelpforum.com/math-he...lume-cone.html