Find the volume of the solid inside the cylinder x^2+y^2 = 2y and bounded above and below by the sphere x^2 + y^2 + z^2 = 4.
Thanks for visiting through my problems. I need to get the solution but i do not have any idea on how to solve it. Please help. Thanks
By completing the square, that cylinder is which has axis (0, 1, z) which is slightly of center from the sphere. I think the first thing I would do is "translate" the origin to (0, 1, 0)- replace y by y'= y- 1 so that y= y'+ 1 and the equation of the cylinder is and the equation of the sphere is or .
Originally Posted by guess
Now the cylinder has its axis at (0, 0, z) and you can use cylindrical coordinates: z will run from the part of the sphere at the bottom, , to the part at the top, while r and range from 0 to 1 and 0 to . The volume is given by
Hi. Here's an alternate way just for fun. Do the complete the square thing, then convert to polar coordinates to obtain , and the sphere into cylindrical coordinates , then the integral is:
which is the same answer as HallsofIvy above, about 9.6 which is in the ball park since the volume of the whole cylinder of length 4 is