Afluid has density 810 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. find the rate of flow outward through the cylinder x2 + y2 = 9, 0 ≤ z ≤ 4.

Denote the cylindrical surface by , and its interior by . By the divergence theorem, the integral of across (the outward flow of the fluid) is equal to the integral of the divergence of over the space it contains, :

The given velocity vector has divergence

Then the total outward flow is

Converting to cylindrical coordinates gives the integral

answer:

c. the third graph