# Practice Problems for Module 14

Covering Sections 10.3, 10.4, and 10.5

Directions.
The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos.
1. Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions:
1. $$\displaystyle 1 < r < 5, \quad 5\pi/3 \leq \theta \leq 7\pi/3$$​

To see the full video page and find related videos, click the following link.

2. $$\displaystyle r=\theta, \quad \theta \geq 0$$​

To see the full video page and find related videos, click the following link.

2. Find a polar equation for the Cartesian equation given by $$xy = 10$$.​

$$\displaystyle r^2=\dfrac{10}{\sin(\theta)\cos(\theta)}$$

To see the full video page and find related videos, click the following link.

3. Find the Cartesian equation for the polar equation given by $$r = 5\sin(\theta)$$.

$$\displaystyle x^2 + \left(y-\frac{5}{2}\right)^2 = \frac{25}{4}$$

To see the full video page and find related videos, click the following link.

4. Find the area of the region $$r = \sqrt{\theta}$$ when $$3\pi/2 \leq \theta \leq 2\pi$$.

$$\displaystyle \dfrac{7\pi^2}{16}$$

To see the full video page and find related videos, click the following link.

5. Find the area of the region $$r = e^{\theta/2}$$ when $$\pi/6 \leq \theta \leq 3\pi/2$$.

$$\displaystyle \dfrac{e^{3\pi/2}-e^{\pi/6}}{2}$$

To see the full video page and find related videos, click the following link.

6. Find the area enclosed by one loop of the curve $$r = \sin(12\theta)$$.

$$\displaystyle \frac{\pi}{48}$$

To see the full video page and find related videos, click the following link.

7. Find the area of the region enclosed by the curve $$r = 3\cos(5\theta)$$.

$$\displaystyle \frac{9\pi}{4}$$

To see the full video page and find related videos, click the following link.

8. Find the area of the region that lies within $$r = 3 - 3\sin(\theta)$$ but outside of the region given by $$r = 3$$.

$$\displaystyle \frac{72-9\pi}{4}$$

To see the full video page and find related videos, click the following link.