Polar Coordinates (College Board AP® Calculus BC): Exam Questions

The polar curves $r=4$ and $r=3-\sin \left(3\theta \right)$ are shown in the graph below for $0\le \theta \le \pi$. Let $R$ be the shaded region inside the graph of $r=4$ and inside the graph of $r=3-\sin \left(3\theta \right)$, as shown.

Find the area of $R$.

For the polar curve $r=4\theta +2\cos \left(4\theta \right)$, find the value of $\frac{dx}{d\theta }$ at $\theta =\frac{\pi }{12}$.

A particle moves along a curve with polar equation $r=4\sin \theta +2e^{-\theta }$ so that $\frac{d\theta }{dt}=2\pi$ for all times $t\ge 0$. Find the value of $\frac{dr}{dt}$ at the point when $\theta =1$.

The graphs of the polar curves $r=2$ and $r=3-2\sin \theta$ are shown below. The curves intersect when $\theta =\frac{\pi }{6}$ and $\theta =\frac{5\pi }{6}$.

Find the area of the shaded region.

Find the slope of the line tangent to the polar curve $r=5-4\cos \theta$ at $\theta =\frac{\pi }{2}$.

The graph of the polar curve $r=4{\theta }^2\sin \left(3\theta \right)$ for $0\le \theta \le \frac{\pi }{3}$ is shown below.

There is a straight line through the origin with positive slope $m$ that divides the area of the region shown into two regions with equal areas.

Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $m$.

The shaded region shown below is bounded by the graph of the polar curve $r=e^{-\frac{\theta }{2}}$ for $0\le \theta \le 2\pi$.

For $k>0$, let $A\left(k\right)$ be the area of the portion of the shaded region that is also inside the circle $r=k\sin \theta$. Find $\underset{k\to \infty }{\lim }A\left(k\right)$.

The figure below shows the polar curves $r=f\left(\theta \right)=1+\sin \theta$ and $r=g\left(\theta \right)=\cos \left(\frac{\theta }{2}\right)$ for $0<\theta <\pi$.

For each $\theta$ in $0<\theta <\pi$, let $h\left(\theta \right)$ be the distance between the points with polar coordinates $\left(f\left(\theta \right),\theta \right)$ and $\left(g\left(\theta \right),\theta \right)$. Write an expression for $h\left(\theta \right)$ and find $h_A$, the average value of $h\left(\theta \right)$ over the interval $0<\theta <\pi$.

Find the value of $\theta$ in the interval $0<\theta <\pi$ for which $h\left(\theta \right)=h_A$. Is the function $h\left(\theta \right)$ increasing or decreasing at this value of $\theta$? Give a reason for your answer.

A particle is moving along the polar curve $r=3+2\cos \theta$ so that at time $t$ seconds, $\theta =t^3$. Find the position vector of the particle in terms of $t$ and find the velocity vector at time $t=0.8$.

For the polar curve $r=\tan \theta$, show that $\frac{dy}{d\theta }=\sin \theta \left(2+{\tan }^2\theta \right)$.

Show that $\frac{d^{2}y}{d{x}^2}=\frac{a-b{\cos }^2\theta }{{\cos }^n\theta }$ where $a$, $b$ and $n$ are positive integers that you should find.

Curve $C$ is defined by the polar equation $r\left(\theta \right)=2{\text{sin}}^2\theta$ for $0\le \theta \le \pi$. Curve $C$ and the semicircle $r=\frac{1}{2}$ for $0\le \theta \le \pi$ are shown in the $xy$-plane.

(Note: Your calculator should be in radian mode.)

Find the rate of change of $r$ with respect to $\theta$ at the point on curve $C$ where $\theta =1.3$. Show the setup for your calculations.

Find the area of the region that lies inside curve $C$ but outside the graph of the polar equation $r=\frac{1}{2}$. Show the setup for your calculations.

It can be shown that $\frac{\text{d}x}{\text{d}\theta }=4\text{sin}\theta {\text{cos}}^2\theta -2{\text{sin}}^3\theta$ for curve $C$. For $0\le \theta \le \frac{\pi }{2}$, find the value of $\theta$ that corresponds to the point on curve $C$ that is farthest from the $y$-axis. Justify your answer.

A particle travels along curve $C$ so that $\frac{\text{d}\theta }{\text{d}t}=15$ for all times $t$. Find the rate at which the particle's distance from the origin changes with respect to time when the particle is at the point where $\theta =1.3$. Show the setup for your calculations.

The polar curves $r=f\left(\theta \right)=1+\text{sin}\theta \text{cos}\left(2\theta \right)$ and $r=g\left(\theta \right)=2\text{cos}\theta$ for $0\le \theta \le \frac{\pi }{2}$ are shown below. Let $R$ be the region in the first quadrant bounded by the curve $r=f\left(\theta \right)$ and the $x$-axis. Let $S$ be the region in the first quadrant bounded by the curve $r=f\left(\theta \right)$, the curve $r=g\left(\theta \right)$, and the $x$-axis.

Find the area of $R$.

The ray $\theta =k$, where $0<k<\frac{\pi }{2}$, divides $S$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $k$.

For each $\theta$ with $0\le \theta \le \frac{\pi }{2}$, let $w\left(\theta \right)$ be the distance between the points with polar coordinates $\left(f\right(\theta ),\theta )$ and $\left(g\right(\theta ),\theta )$. Write an expression for $w\left(\theta \right)$. Find $w_A$, the average value of $w\left(\theta \right)$ over the interval $0\le \theta \le \frac{\pi }{2}$.

Using the information from part (c), find the value of $\theta$ for which $w\left(\theta \right)=w_A$. Is the function $w\left(\theta \right)$ increasing or decreasing at that value of $\theta$? Give a reason for your answer.

Let $S$ be the region bounded by the graph of the polar curve $r\left(\theta \right)=3\sqrt{\theta }\text{sin}\left({\theta }^2\right)$ for $0\le \theta \le \sqrt{\pi }$, as shown in the figure above.

Find the area of $S$.

What is the average distance from the origin to a point on the polar curve $r\left(\theta \right)=3\sqrt{\theta }\text{sin}\left({\theta }^2\right)$ for $0\le \theta \le \sqrt{\pi }$?

There is a line through the origin with positive slope $m$ that divides the region $S$ into two regions with equal areas. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $m$.

For $k>0$, let $A\left(k\right)$ be the area of the portion of region $S$ that is also inside the circle $r=k\text{cos}\theta$. Find $\underset{k\to \infty }{\lim }A\left(k\right)$.

The graphs of the polar curves $r=4$ and $r=3+2\text{cos}\theta$ are shown in the figure below. The curves intersect at $\theta =\frac{\pi }{3}$ and $\theta =\frac{5\pi }{3}$.

Let $R$ be the shaded region that is inside the graph of $r=4$ and also outside the graph of $r=3+2\text{cos}\theta$. Write an expression involving an integral for the area of $R$.

Find the slope of the line tangent to the graph of $r=3+2\text{cos}\theta$ at $\theta =\frac{\pi }{2}$.

A particle moves along the portion of the curve $r=3+2\text{cos}\theta$ for $0<\theta <\frac{\pi }{2}$. The particle moves in such a way that the distance between the particle and the origin increases at a constant rate of $3$ units per second. Find the rate at which the angle $\theta$ changes with respect to time at the instant when the position of the particle corresponds to $\theta =\frac{\pi }{3}$. Indicate units of measure.