Evaluate the surface integral. ∫∫s (x2 + y2 + z2) ds s is the part of the cylinder x2 + y2 = 16 that lies between the planes z = 0 and z = 5, together with its top and bottom disks.

Decompose the surface into three components, , corresponding respectively to the cylindrical region and the top and bottom disks:


where and ,


where and , and


where and .

For the cylinder, we have



and the integral over this surface is






Bottom disk:



and the integral over the bottom disk is





The setup for the integral along the top disk is similar to that for the bottom disk, except that :





Finally, the value of the integral over the entire surface is the sum of the integrals over the component surfaces: