Question
(a) Topic-9.8-Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve
(b) Topic-9.7-Defining Polar Coordinates and Differentiating in Polar Form
(c) Topic-9.7-Defining Polar Coordinates and Differentiating in Polar Form
(d) Topic-9.9-Finding the Area of the Region Bounded by Two Polar Curves
The curve above is drawn in the $xy$-plane and is described by the equation in polar coordinates $r = \theta + \sin(2\theta)$ for $0 \leq \theta \leq \pi$, where $r$ is measured in meters and $\theta$ is measured in radians. The derivative of $r$ with respect to $\theta$ is given by
\(
\frac{dr}{d\theta} = 1 + 2\cos(2\theta).
\)
(a) Find the area bounded by the curve and the $x$-axis.
(b) Find the angle $\theta$ that corresponds to the point on the curve with $x$-coordinate $-2$.
(c) For $\frac{\pi}{3} < \theta < \frac{2\pi}{3}$, $\frac{dr}{d\theta}$ is negative. What does this fact say about $r$? What does this fact say about the curve?
(d) Find the value of $\theta$ in the interval $0 \leq \theta \leq \frac{\pi}{2}$ that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer.
▶️Answer/Explanation
\(\textbf{2(a)}\)
$\text{Area} = \frac{1}{2} \int_0^\pi r^2 \, d\theta$
$= \frac{1}{2} \int_0^\pi \left(\theta + \sin(2\theta)\right)^2 \, d\theta = 4.382$
\(\textbf{2(b)}\)
$-2 = r\cos(\theta) = \left(\theta + \sin(2\theta)\right)\cos(\theta)$
$\theta = 2.786$
\(\textbf{2(c)}\)
Since $\frac{dr}{d\theta} < 0$ for $\frac{\pi}{3} < \theta < \frac{2\pi}{3}$, $r$ is decreasing on this interval. This means the curve is getting closer to the origin.
\(\textbf{2(d)}\)
The only value in $\left[0, \frac{\pi}{2}\right]$ where $\frac{dr}{d\theta} = 0$ is $\theta = \frac{\pi}{3}$.
\( \begin{array}{|c|c|} \hline \theta & r \\ \hline 0 & 0 \\ \frac{\pi}{3} & 1.913 \\ \frac{\pi}{2} & 1.571 \\ \hline \end{array} \)