9.9 Finding the Area of the Region Bounded by Two Polar Curves

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.

Start free

If a region is trapped between two polar curves

{r=f(\theta)}

and

{r=g(\theta)}

on an interval

{[\,\alpha,\beta\,]}

, then its area is

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

where

{f(\theta)}

is the outer curve and

{g(\theta)}

is the inner curve. This is the polar version of area between curves in rectangular coordinates, except the formula uses squared radii because sector area depends on

{r^2}

You should first find where the curves intersect and decide which one lies farther from the origin on the relevant interval. If they switch order, then the region may need to be split into multiple integrals. The most important idea is “outer radius squared minus inner radius squared,” all multiplied by ${\frac{1}{2}}$ . Once the geometry is clear, the setup works just like other area-between-curves problems.

In this unit

9 lessons

Continue the full lesson in Stellar

Create a free account to unlock the remaining sections, checkpoints, course practice, and progress tracking for AP® Calculus BC.

Start free