1. A sphere with radius 2 (http://stuff.daniel15.com/cgi-bin/ma...+y%5E2+z%5E2=4) has its cylindrical core of radius 1 removed (http://stuff.daniel15.com/cgi-bin/ma...?x%5E2+y%5E2=1), 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 http://stuff.daniel15.com/cgi-bin/ma...D%5Cpi%20r%5E2 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 http://stuff.daniel15.com/cgi-bin/ma...pi%20r%5E2%20h 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. http://stuff.daniel15.com/cgi-bin/ma...29j+%28z+x%29k. Verify the Divergence Theorem by showing http://stuff.daniel15.com/cgi-bin/ma...bf%7Bn%7D%20dS where S is the union of the surfaces http://stuff.daniel15.com/cgi-bin/mathtex.cgi?z=0 and http://stuff.daniel15.com/cgi-bin/ma...04-x%5E2-y%5E2 and E is the solid encompassed by those 2 surfaces.

Now I can evaluate the LHS which in spherical coordinates is given by: http://stuff.daniel15.com/cgi-bin/ma...20=%20128%5Cpi (btw is this triple integral right? As in did I set it up right, not the answer http://vcenotes.com/forum/Smileys/alive/tongue.gif)

Now how do I evaluate the RHS for the surface integral over the surface http://stuff.daniel15.com/cgi-bin/ma...04-x%5E2-y%5E2 (without a calculator)... it's almost impossible to do by hand after you compute http://stuff.daniel15.com/cgi-bin/ma...bf%7Br%7D_v%29 where u and v are x and y respectively.

3. Evaluate http://stuff.daniel15.com/cgi-bin/ma...7Bxy%7D%7Ddxdy by making the transformation http://stuff.daniel15.com/cgi-bin/ma...%7Bu%7D%7Bv%7D and http://stuff.daniel15.com/cgi-bin/ma...cgi?y%20=%20uv

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 http://stuff.daniel15.com/cgi-bin/ma...%7B2%7D%7Bv%7D and http://stuff.daniel15.com/cgi-bin/mathtex.cgi?u%20=%201 (I've checked this many times)

Now the Jacobian is http://stuff.daniel15.com/cgi-bin/ma...7B2u%7D%7Bv%7D and after making the transformations we can evaluate the integral in the u-v plane with: http://stuff.daniel15.com/cgi-bin/ma...2ue%5Eu%20dvdu

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?