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 $r=3\sin\theta$

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$\frac{9}{2}{\displaystyle\int }_{0}^{\pi }{\sin}^{2}\theta d\theta$

3. Region in the first quadrant within the cardioid

$r=1+\sin\theta$

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

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$32{\displaystyle\int }_{0}^{\frac{\pi}{2}}{\sin}^{2}\left(2\theta \right)d\theta$

5. Region enclosed by one petal of

$r=\cos\left(3\theta \right)$

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

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$\frac{1}{2}{\displaystyle\int }_{\pi }^{2\pi }{\left(1-\sin\theta \right)}^{2}d\theta$

7. Region in the first quadrant enclosed by

$r=2-\cos\theta$

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

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${\displaystyle\int }_{{\sin}^{-1}\left(\frac{2}{3}\right)}^{\frac{\pi}{2}}{\left(2 - 3\sin\theta \right)}^{2}d\theta$

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

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${\displaystyle\int }_{0}^{\pi }{\left(1 - 2\cos\theta \right)}^{2}d\theta -{\displaystyle\int }_{0}^{\frac{\pi}{3}}{\left(1 - 2\cos\theta \right)}^{2}d\theta$

11. Region common to

$r=3\sin\theta \text{ and }r=2-\sin\theta$

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

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$4{\displaystyle\int }_{0}^{\frac{\pi}{3}}d\theta +16{\displaystyle\int }_{\frac{\pi}{3}}^{\frac{\pi}{2}}\left({\cos}^{2}\theta \right)d\theta$

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 $r=6\sin\theta$

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$9\pi$

15. Above the polar axis enclosed by

$r=2+\sin\theta$

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

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

17. Enclosed by one petal of

$r=4\cos\left(3\theta \right)$

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

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$\frac{9\pi }{8}$

19. Enclosed by

$r=1+\sin\theta$

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

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$\frac{18\pi -27\sqrt{3}}{2}$

21. Enclosed by

$r=2+4\cos\theta$ and outside the inner loop

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

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$\frac{4}{3}\left(4\pi -3\sqrt{3}\right)$

23. Common interior of

$r=3 - 2\sin\theta \text{ and }r=-3+2\sin\theta$

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

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$\frac{3}{2}\left(4\pi -3\sqrt{3}\right)$

25. Inside

$r=1+\cos\theta$ and outside

$r=\cos\theta$

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

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$2\pi -4$

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 $0\le \theta \le 2\pi$

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${\displaystyle\int }_{0}^{2\pi }\sqrt{{\left(1+\sin\theta \right)}^{2}+{\cos}^{2}\theta }d\theta$

29.

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

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

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$\sqrt{2}{\displaystyle\int }_{0}^{1}{e}^{\theta }d\theta$

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

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$\frac{\sqrt{10}}{3}\left({e}^{6}-1\right)$

33.

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

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

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

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37. [T]

$r=\frac{2}{\theta }\text{ on the interval }\pi \le \theta \le 2\pi$

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

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39. [T]

$r=2{\theta }^{2}\text{ on the interval }0\le \theta \le \pi$

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

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

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$A=\pi {\left(\frac{\sqrt{2}}{2}\right)}^{2}=\frac{\pi }{2}\text{ and }\frac{1}{2}{\displaystyle\int }_{0}^{\pi }\left(1+2\sin\theta \cos\theta \right)d\theta =\frac{\pi }{2}$

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

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$C=2\pi \left(\frac{3}{2}\right)=3\pi \text{ and }{\displaystyle\int }_{0}^{\pi }3d\theta =3\pi$

45.

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

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

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$C=2\pi \left(5\right)=10\pi \text{ and }{\displaystyle\int }_{0}^{\pi }10d\theta =10\pi$

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.

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$\frac{dy}{dx}=\frac{{f}^{\prime }\left(\theta \right)\sin\theta +f\left(\theta \right)\cos\theta }{{f}^{\prime }\left(\theta \right)\cos\theta -f\left(\theta \right)\sin\theta }$

49.

$r=1-\sin\theta$ ;

$\left(\frac{1}{2},\frac{\pi }{6}\right)$

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

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The slope is

$\frac{1}{\sqrt{3}}$ .

51.

$r=8\sin\theta$ ;

$\left(4,\frac{5\pi }{6}\right)$

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

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53.

$r=6+3\cos\theta$ ;

$\left(3,\pi \right)$

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

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$\left(4,0\right)$ , the slope is undefined. At

$\left(-4,\frac{\pi }{2}\right)$ , the slope is 0.

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

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

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The slope is undefined at

$\theta =\frac{\pi }{4}$ .

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 $\theta =\frac{\pi }{3}$ .

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

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Slope is

$\frac{-2}{\pi }$ .

61.

$r=\text{ln}\theta$ ,

$\theta =e$

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

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

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Horizontal tangent at

$\left(\pm\sqrt{2},\frac{\pi }{6}\right)$ ,

$\left(\pm\sqrt{2},-\frac{\pi }{6}\right)$ .

65.

$r=2\sin\left(2\theta \right)$

66. The cardioid $r=1+\sin\theta$

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Horizontal tangents at

$\frac{\pi }{2},\frac{7\pi }{6},\frac{11\pi }{6}$ . Vertical tangents at

$\frac{\pi }{6},\frac{5\pi }{6}$ and also at the pole

$\left(0,0\right)$ .

67. Show that the curve

$r=\sin\theta \tan\theta$ (called a cissoid of Diocles) has the line

$x=1$ as a vertical asymptote.

Parametric Equations and Polar Coordinates: Supplemental Content

Problem Set: Area and Arc Length in Polar Coordinates

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