Use the divergence theorem, assuming that the assumptions on T and S are satisfied.

Find the volume of a ball of radius α from Prob. 10.

Data from Prob. 10

Show that a region T with boundary surface S has the volume

where r is the distance of a variable point P: (x, y, z) on S from the origin O and ϕ is the angle between the directed line OP and the outer normal of S at P.

Make a sketch. Use (2) in Sec. 10.7 with F = [x, y, z].