9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Part of Parametric Equations, Polar Coordinates, and Vector-Valued Functions in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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If a region is bounded by a single polar curve

{r=f(\theta)}

from

{\theta=\alpha}

{\theta=\beta}

, then its area is

A=\frac{1}{2}\int_\alpha^\beta [f(\theta)]^2\,d\theta

This formula comes from adding up many tiny sectors rather than vertical or horizontal slices. Because each small sector has area approximately

{\frac{1}{2}r^2\Delta\theta}

, the definite integral gives the exact area in the limit.

You should choose the angle bounds carefully, since they determine which part of the polar graph is being traced. The most common mistake is using the wrong interval for ${\theta}$ , especially for curves that loop or repeat. Once the correct interval is identified, the computation is usually straightforward: square the radius, multiply by ${\frac{1}{2}}$ , and integrate.

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