How to find the volume between the two intersecting surfaces?
$z^2 = x^2 + 3y^2$
$z = 8 - x^2 - y^2$
Please shown the steps needed to solve the problem.
Or show me the direction to go to solve this problem.
Things I tried, that I thought brought me closer to solution.
$x = r \sin(\theta), y = r\cos(\theta)$
we get
$z^2 = r^2 + 2 r^2 \cos^2(\theta) $
and
$z = 8 - r^2 $
equating them I got
$z = {\dfrac{8}{(1 - \sqrt{2 \cos(2 \theta)})}}^{\tfrac{1}{2}}$
I have no idea how to proceed from here, as I cannot find the integration limits.
After curve were plottes, they were smooth, projections appear have a linear profile
