Find the flux of f = x3 i + y3 j + z3 k out of the closed surface z = x2 + y2 , 0 ≤ z ≤ 5 oriented outward. g

The surface is closed, so you can use the divergence theorem:

Then the flux is

where is the region with boundary .

Convert to cylindrical coordinates, taking

Then the integral is

in problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. to find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

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