1. A sphere with radius 2 () has its cylindrical core of radius 1 removed (), what is the volume of the resultant solid?

a) Use cylindrical coordinates to find the volume of the resultant solid.

b) Use spherical coordinates to find the volume of the resultant solid.

For a) I just used cylindrical coordinates to find the volume of the cylinder, then used which is the volume of the sphere then minus the volume of the cylinder from it.

For b) I used spherical coordinates to find the volume of the sphere, then used which is the volume of the cylinder, then used the volume of sphere minus the volume of cylinder.

Now what I am wondering is... is that the right way to interpret the question? Or does the question mean use cylindrical coordinates ONLY to work out the volume? If so... how do I do that? Cause I thought as long as Iusedcylindrical coordinates to find the volume that's fine.

Same goes for b)

2. . Verify the Divergence Theorem by showing where S is the union of the surfaces and and E is the solid encompassed by those 2 surfaces.

Now I can evaluate the LHS which in spherical coordinates is given by: (btw is this triple integral right? As in did I set it up right, not the answer )

Now how do I evaluate the RHS for the surface integral over the surface (without a calculator)... it's almost impossible to do by hand after you compute where u and v are x and y respectively.

3. Evaluate by making the transformation and

I swear you can NOT do this question without a CAS.

After sketching on the u-v plane (with u as the vertical axis) we get the region to be the region between and (I've checked this many times)

Now the Jacobian is and after making the transformations we can evaluate the integral in the u-v plane with:

Now it's impossible to integrate that without a calculator.

Then we can try reverse the order of the integral so we get a dudv instead of dvdu, it's also unable to be integrated by hand (try it yourself). So is there ANYWAY to do this question by hand?