Finding volume bound by unsymmetrical paraboloid

Question

So, going through triple integral volume problems, when establishing the bounds of integration for a solid bound by a paraboloid, one common trick is to convert to cylindrical coordinates allowing for the substitution of x^2+y^2 to r^2. However, what if the coefficient values in front of x and y are different? If for example the equation for the paraboloid that binds the solid is z=x^2/36 + y^2/100? Would I have to set up the triple integral in Cartesian coordinates? I set up my triple integral but it looks exceedingly difficult. How else could I approach this problem.

Answer 1

HINT

Let use

• $x=6r\cos \theta$
• $y=10r\sin \theta$
• $z=z$

calculating the jacobian accordingly.