## Problem Set: Area and Arc Length in Polar Coordinates

For the following exercises, determine a definite integral that represents the area.

1. Region enclosed by $r=4$

2. Region enclosed by $r=3\sin\theta$

3. Region in the first quadrant within the cardioid $r=1+\sin\theta$

4. Region enclosed by one petal of $r=8\sin\left(2\theta \right)$

5. Region enclosed by one petal of $r=\cos\left(3\theta \right)$

6. Region below the polar axis and enclosed by $r=1-\sin\theta$

7. Region in the first quadrant enclosed by $r=2-\cos\theta$

8. Region enclosed by the inner loop of $r=2 - 3\sin\theta$

9. Region enclosed by the inner loop of $r=3 - 4\cos\theta$

10. Region enclosed by $r=1 - 2\cos\theta$ and outside the inner loop

11. Region common to $r=3\sin\theta \text{ and }r=2-\sin\theta$

12. Region common to $r=2\text{ and }r=4\cos\theta$

13. Region common to $r=3\cos\theta \text{ and }r=3\sin\theta$

For the following exercises, find the area of the described region.

14. Enclosed by $r=6\sin\theta$

15. Above the polar axis enclosed by $r=2+\sin\theta$

16. Below the polar axis and enclosed by $r=2-\cos\theta$

17. Enclosed by one petal of $r=4\cos\left(3\theta \right)$

18. Enclosed by one petal of $r=3\cos\left(2\theta \right)$

19. Enclosed by $r=1+\sin\theta$

20. Enclosed by the inner loop of $r=3+6\cos\theta$

21. Enclosed by $r=2+4\cos\theta$ and outside the inner loop

22. Common interior of $r=4\sin\left(2\theta \right)\text{and }r=2$

23. Common interior of $r=3 - 2\sin\theta \text{ and }r=-3+2\sin\theta$

24. Common interior of $r=6\sin\theta \text{ and }r=3$

25. Inside $r=1+\cos\theta$ and outside $r=\cos\theta$

26. Common interior of $r=2+2\cos\theta \text{ and }r=2\sin\theta$

For the following exercises, find a definite integral that represents the arc length.

27. $r=4\cos\theta \text{ on the interval }0\le \theta \le \frac{\pi }{2}$

28. $r=1+\sin\theta$ on the interval $0\le \theta \le 2\pi$

29. $r=2\sec\theta \text{ on the interval }0\le \theta \le \frac{\pi }{3}$

30. $r={e}^{\theta }\text{ on the interval }0\le \theta \le 1$

For the following exercises, find the length of the curve over the given interval.

31. $r=6\text{ on the interval }0\le \theta \le \frac{\pi }{2}$

32. $r={e}^{3\theta }\text{ on the interval }0\le \theta \le 2$

33. $r=6\cos\theta \text{ on the interval }0\le \theta \le \frac{\pi }{2}$

34. $r=8+8\cos\theta \text{ on the interval }0\le \theta \le \pi$

35. $r=1-\sin\theta \text{ on the interval }0\le \theta \le 2\pi$

For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.

36. [T] $r=3\theta \text{ on the interval }0\le \theta \le \frac{\pi }{2}$

37. [T] $r=\frac{2}{\theta }\text{ on the interval }\pi \le \theta \le 2\pi$

38. [T] $r={\sin}^{2}\left(\frac{\theta }{2}\right)\text{ on the interval }0\le \theta \le \pi$

39. [T] $r=2{\theta }^{2}\text{ on the interval }0\le \theta \le \pi$

40. [T] $r=\sin\left(3\cos\theta \right)\text{ on the interval }0\le \theta \le \pi$

For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.

41. $r=3\sin\theta \text{ on the interval }0\le \theta \le \pi$

42. $r=\sin\theta +\cos\theta \text{ on the interval }0\le \theta \le \pi$

43. $r=6\sin\theta +8\cos\theta \text{ on the interval }0\le \theta \le \pi$

For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.

44. $r=3\sin\theta \text{ on the interval }0\le \theta \le \pi$

45. $r=\sin\theta +\cos\theta \text{ on the interval }0\le \theta \le \pi$

46. $r=6\sin\theta +8\cos\theta \text{ on the interval }0\le \theta \le \pi$

47. Verify that if $y=r\sin\theta =f\left(\theta \right)\sin\theta$ then $\frac{dy}{d\theta }=f\prime \left(\theta \right)\sin\theta +f\left(\theta \right)\cos\theta$.

For the following exercises, find the slope of a tangent line to a polar curve $r=f\left(\theta \right)$. Let $x=r\cos\theta =f\left(\theta \right)\cos\theta$ and $y=r\sin\theta =f\left(\theta \right)\sin\theta$, so the polar equation $r=f\left(\theta \right)$ is now written in parametric form.

48. Use the definition of the derivative $\frac{dy}{dx}=\frac{\frac{dy}{d}\theta }{\frac{dx}{d}\theta }$ and the product rule to derive the derivative of a polar equation.

49. $r=1-\sin\theta$; $\left(\frac{1}{2},\frac{\pi }{6}\right)$

50. $r=4\cos\theta$; $\left(2,\frac{\pi }{3}\right)$

51. $r=8\sin\theta$; $\left(4,\frac{5\pi }{6}\right)$

52. $r=4+\sin\theta$; $\left(3,\frac{3\pi }{2}\right)$

53. $r=6+3\cos\theta$; $\left(3,\pi \right)$

54. $r=4\cos\left(2\theta \right)$; tips of the leaves

55. $r=2\sin\left(3\theta \right)$; tips of the leaves

56. $r=2\theta$; $\left(\frac{\pi }{2},\frac{\pi }{4}\right)$

57. Find the points on the interval $\text{-}\pi \le \theta \le \pi$ at which the cardioid $r=1-\cos\theta$ has a vertical or horizontal tangent line.

58. For the cardioid $r=1+\sin\theta$, find the slope of the tangent line when $\theta =\frac{\pi }{3}$.

For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of $\theta$.

59. $r=3\cos\theta ,\theta =\frac{\pi }{3}$

60. $r=\theta$, $\theta =\frac{\pi }{2}$

61. $r=\text{ln}\theta$, $\theta =e$

62. [T] Use technology: $r=2+4\cos\theta$ at $\theta =\frac{\pi }{6}$

For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line.

63. $r=4\cos\theta$

64. ${r}^{2}=4\cos\left(2\theta \right)$

65. $r=2\sin\left(2\theta \right)$

66. The cardioid $r=1+\sin\theta$

67. Show that the curve $r=\sin\theta \tan\theta$ (called a cissoid of Diocles) has the line $x=1$ as a vertical asymptote.