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 Incoming links: Flux of the position vector through a cone

Compute the flux directly using the outward vector area element, as required. Restrict the vector field to the surface parametric equation and take the dot product:

F (R) \cdot d S = [\frac{b}{a} r^{2} (cos^{2} θ + sin^{2} θ) + b r (1 - \frac{r}{a})] d r d θ = b r d r d θ .

The terms quadratic in

r

cancel. Therefore the surface integral is

\int_{C} F \cdot d S = \int_{0}^{2 π} \int_{0}^{a} b r d r d θ = π a^{2} b .