 Polar area formula

ID: polar-area-formula

Suppose

0 \leq r_{1} (θ) \leq r_{2} (θ)

and the region is described without overlap by

α \leq θ \leq β

and

r_{1} (θ) \leq r \leq r_{2} (θ)

. Integrating the polar area element gives its area:

A = \int_{α}^{β} \int_{r_{1} (θ)}^{r_{2} (θ)} r d r d θ = \frac{1}{2} \int_{α}^{β} (r_{2} (θ)^{2} - r_{1} (θ)^{2}) d θ .

For a region bounded by a single radial graph and two rays, take

r_{1} = 0

. If the two bounding graphs cross, split the angular interval at their intersections so that the larger radius is used consistently.

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