Learning Outcomes
- Compute areas bounded by curves given in polar coordinates.
Section 7.6 Polar Area (CO6)
Subsection 7.6.1 Activities
Activity 7.6.1.
Consider the regions bounded by polar coordinates \(0\leq r \leq r_0\) and \(0\leq \theta \leq \theta_0\) for various values of \(r_0, \theta_0\text{.}\)
(a)
Sketch the region bound by polar coordinates \(0\leq r \leq 1\) and \(0\leq \theta \leq \dfrac{\pi}{3}\text{.}\) What is the area of this region?
(b)
Sketch the region bound by polar coordinates \(0\leq r \leq 2\) and \(0\leq \theta \leq \dfrac{\pi}{3}\text{.}\) What is the area of this region?
(c)
Sketch the region bound by polar coordinates \(0\leq r \leq 5\) and \(0\leq \theta \leq \dfrac{\pi}{3}\text{.}\) What is the area of this region?
(d)
Sketch the region bound by polar coordinates \(0\leq r \leq 1\) and \(0\leq \theta \leq \dfrac{\pi}{4}\text{.}\) What is the area of this region?
(e)
Sketch the region bound by polar coordinates \(0\leq r \leq 5\) and \(0\leq \theta \leq \dfrac{\pi}{4}\text{.}\) What is the area of this region?
Activity 7.6.2.
What in general is the area of the region bound by polar coordinates \(0\leq r \leq r_0\) and \(0\leq \theta \leq \theta_0\text{?}\)
- \(\displaystyle \displaystyle \pi \frac{r_0^2}{\theta_0}\)
- \(\displaystyle \displaystyle \frac{r_0^2}{\pi \theta_0}\)
- \(\displaystyle \displaystyle \theta_0 \frac{r_0^2}{\pi}\)
- \(\displaystyle \displaystyle \theta\frac{r_0^2}{2}\)
Activity 7.6.3.
Consider the “fan-shaped” region between the pole and \(r=f(\theta)\) as the angle \(\theta\) ranges from \(\alpha\) to \(\beta\) as depicted in Figure 179.
(a)
Which of the following best describes a Riemann sum which approximates the area of this region?
- \(\displaystyle \displaystyle \sum_{k=1}^n f(\theta_k)\Delta \theta\)
- \(\displaystyle \displaystyle \sum_{k=1}^n f(\theta_k)^2\Delta \theta\)
- \(\displaystyle \displaystyle \sum_{k=1}^n \frac{f(\theta_k)^2}{2}\Delta \theta\)
- \(\displaystyle \displaystyle \sum_{k=1}^n \pi f(\theta_k)^2\Delta \theta\)
(b)
Which of the following describes an integral which computes the area of this region?
- \(\displaystyle \displaystyle \int_{\theta=\alpha}^{\theta=\beta} f(\theta)d\theta\)
- \(\displaystyle \displaystyle \int_{\theta=\alpha}^{\theta=\beta}f(\theta)^2 d\theta\)
- \(\displaystyle \displaystyle \int_{\theta=\alpha}^{\theta=\beta} \frac{f(\theta)^2}{2}d\theta\)
- \(\displaystyle \displaystyle \int_{\theta=\alpha}^{\theta=\beta} \pi f(\theta)^2 d\theta\)
Fact 7.6.4.
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*}
Activity 7.6.5.
(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 \(r=\cos(2\theta)\text{.}\)
Subsection 7.6.2 Videos
Subsection 7.6.3 Exercises
Exercises available at
https://tbil.org/preview/calculus/exercises/#/bank/CO6/
