Flux through a paraboloid.

multivariable-calculus

Solution 1:

The problem is when you do $$\vec{F}\cdot \vec{n}$$ Your integral should become $$\iint \vec{F}\cdot\vec{n}dS =\iint\vec{F}\cdot \frac{\nabla \vec{f}}{||\nabla \vec{f}||} ||\nabla \vec{f}|| dA =\iint 4(1-z)dA$$

You should not have the denominator when you use polar coordinate.

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