Divergence Theorem, Find the outward flux of the vector field F...?

Question

Find the outward flux of the vector field F = (x^3, y^3,z^2) across the surface of the region that is enclosed by the circular cylinder x^2 + y^2 = 36 and the planes z = 0 and z = 2.

Don't know how to do this

Answer 1

The title of your question gives the answer: by the Divergence Theorem, the outward flux of F across the surface equals the (triple) integral of the divergence of F over the volume enclosed by the surface.

Flux = ∫∫∫ div F dV = ∫∫∫ div F dz dx dy

I would do the simple z-integration first (innermost integral); the limits are clearly z=0 to z=2. The x and y integrations amounts to integrating over the circle x² + y² = 36, so if you do x-integration next, the limits are from -√(36 - y²) to √(36 - y²), and then y goes from -6. to 6.