Question

1a) suppose we want to integrate a funciton f(x,y) over the ellipse x^2/a^2 + y^2/b^2=1 (where a is not equal to b are positive constants). Suggest a suitable change of variables, and find the Jacobian of this transformation. (use a modified version of polar coordinates)

b) Using the answer to (a), find the volume of the region enclosed beneath the elliptic paraboloid z = 12-x^2-3y^2 and above the xy-plane.