Calculus 3 Challeng problems

1. Let f(x,y) = 3$x^{2} $ -10xy+3$y^{2} $ on the region [0,4] x [0,4]; evaluate the double integral 

2. Change the region to 0 < x < 3 and 0 < y < $\sqrt{9-x^2}$ ; evaluate the double integral of the same function, but in polar coordinates.

3. Let x = 3u + v and y = u + 3v; find the Jacobian transform J(u,v), then evaluate the double integral on the corresponding uv-region to that given iproblem 1.

4. Let f(x,y,z) = 5$x^{4} $ +10$x^{2} $$y^{2} $ + 5$y^{4} $ + 5$x^{2} $$z^{2} $ + 5$y^{2}$$z^{2} $  on the first octant of a sphere of radius 4; evaluate the triple integral 

5. Let x =3w$\sqrt{u+v} $ , y =3w$\sqrt{u-v} $ and z = 3w; Find the Jacobian transform J(u,v,w), then evaluate the triple integral on the corresponding soild to that in problem 4.