[SOLVED] Calculate a volume + theoretical question
1)Identify the region of integration and calculate the following integral using spherical coordinates : .
My attempt : I notice that the integrand is not the Jacobian of the change of variable spherical to Cartesian but almost. I mean that instead of . Thus I am at a loss.
I don't know how to answer the question.
2) I'd like to know a general method (i.e. applicable in all cases) that transform a complicated solid (like a cone, a paraboloid, hyperboloid or anything different from a cylinder, cube and sphere) into a cube. I'd appreciate any book reference or an example with the above exercise.
I ask this because even though it might be difficult to find such a transformation, if I success in finding it then its Jacobian shouldn't be a problem to find and so the volume of the complicated solid should be quite easy to solve.