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Section 7.6 Polar area (CO6)
Learning Outcomes
Fact 7.6.1.
The area of the “fan-shaped” region between the pole and \(r=f(\theta)\) as the angle \(\theta\) ranges from \(\alpha\) to \(\beta\) is given by
\begin{equation*}
\int_{\theta=\alpha}^{\theta=\beta} \frac{r^2}{2}d\theta\text{.}
\end{equation*}
Figure 148. Finding the polar area differential
Activity 7.6.2.
(a)
Find an integral computing the area of the region defined by \(0\leq r\leq-\cos(\theta)+5\) and \(\pi/2\leq \theta\leq 3\pi/4\text{.}\)
(b)
Find the area enclosed by the cardioid \(r=2(1+\cos(\theta)\text{.}\)
(c)
Find the area enclosed by one loop of the 4-petaled rose \(r=\cos(2\theta)\text{.}\)
