For the following exercises, determine a definite integral that represents the area.
1. Region enclosed by [latex]r=4[/latex]
2. Region enclosed by [latex]r=3\sin\theta[/latex]
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[latex]\frac{9}{2}{\displaystyle\int }_{0}^{\pi }{\sin}^{2}\theta d\theta[/latex]
3. Region in the first quadrant within the cardioid [latex]r=1+\sin\theta[/latex]
4. Region enclosed by one petal of [latex]r=8\sin\left(2\theta \right)[/latex]
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[latex]32{\displaystyle\int }_{0}^{\frac{\pi}{2}}{\sin}^{2}\left(2\theta \right)d\theta[/latex]
5. Region enclosed by one petal of [latex]r=\cos\left(3\theta \right)[/latex]
6. Region below the polar axis and enclosed by [latex]r=1-\sin\theta[/latex]
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[latex]\frac{1}{2}{\displaystyle\int }_{\pi }^{2\pi }{\left(1-\sin\theta \right)}^{2}d\theta[/latex]
7. Region in the first quadrant enclosed by [latex]r=2-\cos\theta[/latex]
8. Region enclosed by the inner loop of [latex]r=2 - 3\sin\theta[/latex]
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[latex]{\displaystyle\int }_{{\sin}^{-1}\left(\frac{2}{3}\right)}^{\frac{\pi}{2}}{\left(2 - 3\sin\theta \right)}^{2}d\theta[/latex]
9. Region enclosed by the inner loop of [latex]r=3 - 4\cos\theta[/latex]
10. Region enclosed by [latex]r=1 - 2\cos\theta[/latex] and outside the inner loop
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[latex]{\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[/latex]
11. Region common to [latex]r=3\sin\theta \text{ and }r=2-\sin\theta[/latex]
12. Region common to [latex]r=2\text{ and }r=4\cos\theta[/latex]
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[latex]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[/latex]
13. Region common to [latex]r=3\cos\theta \text{ and }r=3\sin\theta[/latex]
For the following exercises, find the area of the described region.
14. Enclosed by [latex]r=6\sin\theta[/latex]
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[latex]9\pi[/latex]
15. Above the polar axis enclosed by [latex]r=2+\sin\theta[/latex]
16. Below the polar axis and enclosed by [latex]r=2-\cos\theta[/latex]
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[latex]\frac{9\pi }{4}[/latex]
17. Enclosed by one petal of [latex]r=4\cos\left(3\theta \right)[/latex]
18. Enclosed by one petal of [latex]r=3\cos\left(2\theta \right)[/latex]
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[latex]\frac{9\pi }{8}[/latex]
19. Enclosed by [latex]r=1+\sin\theta[/latex]
20. Enclosed by the inner loop of [latex]r=3+6\cos\theta[/latex]
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[latex]\frac{18\pi -27\sqrt{3}}{2}[/latex]
21. Enclosed by [latex]r=2+4\cos\theta[/latex] and outside the inner loop
22. Common interior of [latex]r=4\sin\left(2\theta \right)\text{and }r=2[/latex]
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[latex]\frac{4}{3}\left(4\pi -3\sqrt{3}\right)[/latex]
23. Common interior of [latex]r=3 - 2\sin\theta \text{ and }r=-3+2\sin\theta[/latex]
24. Common interior of [latex]r=6\sin\theta \text{ and }r=3[/latex]
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[latex]\frac{3}{2}\left(4\pi -3\sqrt{3}\right)[/latex]
25. Inside [latex]r=1+\cos\theta[/latex] and outside [latex]r=\cos\theta[/latex]
26. Common interior of [latex]r=2+2\cos\theta \text{ and }r=2\sin\theta[/latex]
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[latex]2\pi -4[/latex]
For the following exercises, find a definite integral that represents the arc length.
27. [latex]r=4\cos\theta \text{ on the interval }0\le \theta \le \frac{\pi }{2}[/latex]
28. [latex]r=1+\sin\theta[/latex] on the interval [latex]0\le \theta \le 2\pi[/latex]
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[latex]{\displaystyle\int }_{0}^{2\pi }\sqrt{{\left(1+\sin\theta \right)}^{2}+{\cos}^{2}\theta }d\theta[/latex]
29. [latex]r=2\sec\theta \text{ on the interval }0\le \theta \le \frac{\pi }{3}[/latex]
30. [latex]r={e}^{\theta }\text{ on the interval }0\le \theta \le 1[/latex]
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[latex]\sqrt{2}{\displaystyle\int }_{0}^{1}{e}^{\theta }d\theta[/latex]
For the following exercises, find the length of the curve over the given interval.
31. [latex]r=6\text{ on the interval }0\le \theta \le \frac{\pi }{2}[/latex]
32. [latex]r={e}^{3\theta }\text{ on the interval }0\le \theta \le 2[/latex]
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[latex]\frac{\sqrt{10}}{3}\left({e}^{6}-1\right)[/latex]
33. [latex]r=6\cos\theta \text{ on the interval }0\le \theta \le \frac{\pi }{2}[/latex]
34. [latex]r=8+8\cos\theta \text{ on the interval }0\le \theta \le \pi[/latex]
35. [latex]r=1-\sin\theta \text{ on the interval }0\le \theta \le 2\pi[/latex]
For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.
36. [T] [latex]r=3\theta \text{ on the interval }0\le \theta \le \frac{\pi }{2}[/latex]
37. [T] [latex]r=\frac{2}{\theta }\text{ on the interval }\pi \le \theta \le 2\pi[/latex]
38. [T] [latex]r={\sin}^{2}\left(\frac{\theta }{2}\right)\text{ on the interval }0\le \theta \le \pi[/latex]
39. [T] [latex]r=2{\theta }^{2}\text{ on the interval }0\le \theta \le \pi[/latex]
40. [T] [latex]r=\sin\left(3\cos\theta \right)\text{ on the interval }0\le \theta \le \pi[/latex]
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. [latex]r=3\sin\theta \text{ on the interval }0\le \theta \le \pi[/latex]
42. [latex]r=\sin\theta +\cos\theta \text{ on the interval }0\le \theta \le \pi[/latex]
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[latex]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}[/latex]
43. [latex]r=6\sin\theta +8\cos\theta \text{ on the interval }0\le \theta \le \pi[/latex]
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. [latex]r=3\sin\theta \text{ on the interval }0\le \theta \le \pi[/latex]
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[latex]C=2\pi \left(\frac{3}{2}\right)=3\pi \text{ and }{\displaystyle\int }_{0}^{\pi }3d\theta =3\pi[/latex]
45. [latex]r=\sin\theta +\cos\theta \text{ on the interval }0\le \theta \le \pi[/latex]
46. [latex]r=6\sin\theta +8\cos\theta \text{ on the interval }0\le \theta \le \pi[/latex]
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[latex]C=2\pi \left(5\right)=10\pi \text{ and }{\displaystyle\int }_{0}^{\pi }10d\theta =10\pi[/latex]
47. Verify that if [latex]y=r\sin\theta =f\left(\theta \right)\sin\theta[/latex] then [latex]\frac{dy}{d\theta }=f\prime \left(\theta \right)\sin\theta +f\left(\theta \right)\cos\theta[/latex].
For the following exercises, find the slope of a tangent line to a polar curve [latex]r=f\left(\theta \right)[/latex]. Let [latex]x=r\cos\theta =f\left(\theta \right)\cos\theta[/latex] and [latex]y=r\sin\theta =f\left(\theta \right)\sin\theta[/latex], so the polar equation [latex]r=f\left(\theta \right)[/latex] is now written in parametric form.
48. Use the definition of the derivative [latex]\frac{dy}{dx}=\frac{\frac{dy}{d}\theta }{\frac{dx}{d}\theta }[/latex] and the product rule to derive the derivative of a polar equation.
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[latex]\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 }[/latex]
49. [latex]r=1-\sin\theta[/latex]; [latex]\left(\frac{1}{2},\frac{\pi }{6}\right)[/latex]
50. [latex]r=4\cos\theta[/latex]; [latex]\left(2,\frac{\pi }{3}\right)[/latex]
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The slope is [latex]\frac{1}{\sqrt{3}}[/latex].
51. [latex]r=8\sin\theta[/latex]; [latex]\left(4,\frac{5\pi }{6}\right)[/latex]
52. [latex]r=4+\sin\theta[/latex]; [latex]\left(3,\frac{3\pi }{2}\right)[/latex]
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The slope is 0.
53. [latex]r=6+3\cos\theta[/latex]; [latex]\left(3,\pi \right)[/latex]
54. [latex]r=4\cos\left(2\theta \right)[/latex]; tips of the leaves
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At [latex]\left(4,0\right)[/latex], the slope is undefined. At [latex]\left(-4,\frac{\pi }{2}\right)[/latex], the slope is 0.
55. [latex]r=2\sin\left(3\theta \right)[/latex]; tips of the leaves
56. [latex]r=2\theta[/latex]; [latex]\left(\frac{\pi }{2},\frac{\pi }{4}\right)[/latex]
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The slope is undefined at [latex]\theta =\frac{\pi }{4}[/latex].
57. Find the points on the interval [latex]\text{-}\pi \le \theta \le \pi[/latex] at which the cardioid [latex]r=1-\cos\theta[/latex] has a vertical or horizontal tangent line.
58. For the cardioid [latex]r=1+\sin\theta[/latex], find the slope of the tangent line when [latex]\theta =\frac{\pi }{3}[/latex].
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Slope = −1.
For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of [latex]\theta[/latex].
59. [latex]r=3\cos\theta ,\theta =\frac{\pi }{3}[/latex]
60. [latex]r=\theta[/latex], [latex]\theta =\frac{\pi }{2}[/latex]
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Slope is [latex]\frac{-2}{\pi }[/latex].
61. [latex]r=\text{ln}\theta[/latex], [latex]\theta =e[/latex]
62. [T] Use technology: [latex]r=2+4\cos\theta[/latex] at [latex]\theta =\frac{\pi }{6}[/latex]
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Calculator answer: −0.836.
For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line.
63. [latex]r=4\cos\theta[/latex]
64. [latex]{r}^{2}=4\cos\left(2\theta \right)[/latex]
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Horizontal tangent at [latex]\left(\pm\sqrt{2},\frac{\pi }{6}\right)[/latex], [latex]\left(\pm\sqrt{2},-\frac{\pi }{6}\right)[/latex].
65. [latex]r=2\sin\left(2\theta \right)[/latex]
66. The cardioid [latex]r=1+\sin\theta[/latex]
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Horizontal tangents at [latex]\frac{\pi }{2},\frac{7\pi }{6},\frac{11\pi }{6}[/latex]. Vertical tangents at [latex]\frac{\pi }{6},\frac{5\pi }{6}[/latex] and also at the pole [latex]\left(0,0\right)[/latex].
67. Show that the curve [latex]r=\sin\theta \tan\theta[/latex] (called a cissoid of Diocles) has the line [latex]x=1[/latex] as a vertical asymptote.
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