AP Calculus BC / Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.8 Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve
Notes
$$A=\frac{1}{2}\int_\alpha^\beta r^2\,d\theta$$
- Find the $\theta$-values that trace the region exactly once.
Practice
Area inside $r=2$:
$\displaystyle \frac{1}{2}\int_0^{2\pi}4\,d\theta$.
The area inside one petal of $r=\sin(2\theta)$ uses $\theta$ from:
$r\ge0$ between consecutive zeros at 0 and $\displaystyle \frac{\pi}{2}$.
Area of that petal:
$\displaystyle \frac{1}{2}\int_0^{\pi/2}\sin^2(2\theta)d\theta=\frac{1}{2}\cdot\frac{\pi}{4}$.
Ready for more? Take the Unit 9 test.