Find the volumes of each of the following by computing volume integral.?

Question

1.) The cylinder of radius 1 and height 3 with its base is centered at the origin of the xy-plane and its axis of symmetry along the positive z-axis.

2.) The upper hemisphere of the sphere of radius 2 centered at the origin.

Answer 1

1.) You want S[0,3]S[-1,1]S[-1,1](x^2 + y^2)dxdydz, where S[a,b] means integrate from a to b. Of course you know how to check your answer using the standard formula for area of a right circular cylinder.

2.) You want S[0,2]S[-2,2]S[-2,2](x^2 + y^2 + z^2)dxdydz.