Concept preview

What you will be able to do

1. 01State and apply the polar area formula $A=\dfrac{1}{2}\displaystyle\int_\alpha^\beta r^2\,d\theta$, understanding why the factor $\tfrac{1}{2}$ is essential.
2. 02Determine the correct angular limits $\alpha$ and $\beta$ for one petal, one loop, or a full traversal of a polar curve.
3. 03Compute the area enclosed by a circle, a cardioid, and a single rose petal, keeping answers exact with $\pi$.

Recall the area of a circular sector: a slice of a disk of radius $r$ subtending a central angle $\Delta\theta$ has area $\tfrac{1}{2}r^2\,\Delta\theta$. (Check: a full turn $\Delta\theta=2\pi$ gives $\tfrac{1}{2}r^2\cdot2\pi=\pi r^2$, the whole disk.) Now take a polar region bounded by $r=f(\theta)$ and the two rays $\theta=\alpha$ and $\theta=\beta$. Chop the angular interval into many tiny wedges of width $d\theta$. Over one such wedge the radius is nearly constant at $r=f(\theta)$, so the wedge is almost a sector of area $dA=\tfrac{1}{2}r^2\,d\theta$.