Stewart Section 10.4: Areas and Lengths in Polar Coordinates

1 Polar Area · Level 2

Find the area of the region that is bounded by the given curve and lies in the specified sector. \(r = e^{-\dfrac{\theta}{4}}\), \(\dfrac{\pi}{2} \leq \theta \leq \pi\)

2 Polar Area · Level 2

Find the area of the region that is bounded by the given curve and lies in the specified sector. \(r = \cos \theta\), \(0 \leq \theta \leq \dfrac{\pi}{6}\)

3 Polar Area · Level 2

Find the area of the region that is bounded by the given curve and lies in the specified sector. \(r = \sin \theta + \cos \theta\), \(0 \leq \theta \leq \pi\)

4 Polar Area · Level 2

Find the area of the region that is bounded by the given curve and lies in the specified sector. \(r = \dfrac{1}{\theta}\), \(\dfrac{\pi}{2} \leq \theta \leq 2 \pi\)

5 Polar Area · Level 3

Find the area of the shaded region. \(r^2 = \sin 2 \theta\)

6 Polar Area · Level 3

Find the area of the shaded region. \(r = 4 + 3 \sin \theta\)

7 Polar Area · Level 3

Find the area of the shaded region.

8 Polar Area · Level 3

Find the area of the shaded region. \(r = \sqrt{\ln \theta}\), \(1 \leq \theta \leq 2 \pi\)

9 Polar Area · Level 2

Sketch the curve and find the area that it encloses. \(r = 2 \sin \theta\)

10 Polar Area · Level 2

Sketch the curve and find the area that it encloses. \(r = 1 - \sin \theta\)

11 Polar Area · Level 2

Sketch the curve and find the area that it encloses. \(r = 3 + 2 \cos \theta\)

12 Polar Area · Level 2

Sketch the curve and find the area that it encloses. \(r = 2 - \cos \theta\)

13 Polar Area · Level 3

Graph the curve and find the area that it encloses. \(r = 2 + \sin 4 \theta\)

14 Polar Area · Level 3

Graph the curve and find the area that it encloses. \(r = 3 - 2 \cos 4 \theta\)

15 Polar Area · Level 3

Graph the curve and find the area that it encloses. \(r = \sqrt{1 + \cos^2 (5 \theta)}\)

16 Polar Area · Level 3

Graph the curve and find the area that it encloses. \(r = 1 + 5 \sin 6 \theta\)

17 Polar Area - One Loop · Level 3

Find the area of the region enclosed by one loop of the curve. \(r = 4 \cos 3 \theta\)

18 Polar Area - One Loop · Level 3

Find the area of the region enclosed by one loop of the curve. \(r^2 = 4 \cos 2 \theta\)

19 Polar Area - One Loop · Level 3

Find the area of the region enclosed by one loop of the curve. \(r = \sin 4 \theta\)

20 Polar Area - One Loop · Level 3

Find the area of the region enclosed by one loop of the curve. \(r = 2 \sin 5 \theta\)

21 Polar Area - Inner Loop · Level 3

Find the area of the region enclosed by one loop of the curve. \(r = 1 + 2 \sin \theta\) (inner loop)

22 Polar Area · Level 4

Find the area enclosed by the loop of the strophoid \(r = 2 \cos \theta - \sec \theta\).

23 Polar Area - Between Curves · Level 3

Find the area of the region that lies inside the first curve and outside the second curve. \(r = 4 \sin \theta\), \(r = 2\)

24 Polar Area - Between Curves · Level 3

Find the area of the region that lies inside the first curve and outside the second curve. \(r = 1 - \sin \theta\), \(r = 1\)

25 Polar Area - Between Curves · Level 3

Find the area of the region that lies inside the first curve and outside the second curve. \(r^2 = 8 \cos 2 \theta\), \(r = 2\)

26 Polar Area - Between Curves · Level 3

Find the area of the region that lies inside the first curve and outside the second curve. \(r = 1 + \cos \theta\), \(r = 2 - \cos \theta\)

27 Polar Area - Between Curves · Level 3

Find the area of the region that lies inside the first curve and outside the second curve. \(r = 3 \cos \theta\), \(r = 1 + \cos \theta\)

28 Polar Area - Between Curves · Level 3

Find the area of the region that lies inside the first curve and outside the second curve. \(r = 3 \sin \theta\), \(r = 2 - \sin \theta\)

29 Polar Area - Intersection · Level 3

Find the area of the region that lies inside both curves. \(r = 3 \sin \theta\), \(r = 3 \cos \theta\)

30 Polar Area - Intersection · Level 3

Find the area of the region that lies inside both curves. \(r = 1 + \cos \theta\), \(r = 1 - \cos \theta\)

31 Polar Area - Intersection · Level 3

Find the area of the region that lies inside both curves. \(r = \sin 2 \theta\), \(r = \cos 2 \theta\)

32 Polar Area - Intersection · Level 3

Find the area of the region that lies inside both curves. \(r = 3 + 2 \cos \theta\), \(r = 3 + 2 \sin \theta\)

33 Polar Area - Intersection · Level 3

Find the area of the region that lies inside both curves. \(r^2 = 2 \sin 2 \theta\), \(r = 1\)

34 Polar Area - Intersection · Level 4

Find the area of the region that lies inside both curves. \(r = a \sin \theta\), \(r = b \cos \theta\), \(a > 0\), \(b > 0\)

35 Polar Area - Limacon · Level 4

Find the area inside the larger loop and outside the smaller loop of the limacon \(r = \dfrac{1}{2} + \cos \theta\).

36 Polar Area · Level 4

Find the area between a large loop and the enclosed small loop of the curve \(r = 1 + 2 \cos 3 \theta\).

37 Polar Curves - Intersection Points · Level 3

Find all points of intersection of the given curves. \(r = \sin \theta\), \(r = 1 - \sin \theta\)

38 Polar Curves - Intersection Points · Level 3

Find all points of intersection of the given curves. \(r = 1 + \cos \theta\), \(r = 1 - \sin \theta\)

39 Polar Curves - Intersection Points · Level 3

Find all points of intersection of the given curves. \(r = 2 \sin 2 \theta\), \(r = 1\)

40 Polar Curves - Intersection Points · Level 3

Find all points of intersection of the given curves. \(r = \cos 3 \theta\), \(r = \sin 3 \theta\)

41 Polar Curves - Intersection Points · Level 3

Find all points of intersection of the given curves. \(r = \sin \theta\), \(r = \sin 2 \theta\)

42 Polar Curves - Intersection Points · Level 3

Find all points of intersection of the given curves. \(r^2 = \sin 2 \theta\), \(r^2 = \cos 2 \theta\)

43 Polar Area - Numerical · Level 3

The points of intersection of the cardioid \(r = 1 + \sin \theta\) and the spiral loop \(r = 2 \theta\), \(-\dfrac{\pi}{2} \leq \theta \leq \dfrac{\pi}{2}\), can't be found exactly. Use a graphing device to find the approximate values of \(\theta\) at which they intersect. Then use these values to estimate the area that lies inside both curves.

44 Polar Area - Application · Level 3

When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 4 m from the front of the stage (as in the figure) and the boundary of the optimal pickup region is given by the cardioid \(r = 8 + 8 \sin \theta\), where \(r\) is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microphone. Answer their question.

45 Polar Arc Length · Level 3

Find the exact length of the polar curve. \(r = 2 \cos \theta\), \(0 \leq \theta \leq \pi\)

46 Polar Arc Length · Level 3

Find the exact length of the polar curve. \(r = 5^\theta\), \(0 \leq \theta \leq 2 \pi\)

47 Polar Arc Length · Level 3

Find the exact length of the polar curve. \(r = \theta^2\), \(0 \leq \theta \leq 2 \pi\)

48 Polar Arc Length · Level 3

Find the exact length of the polar curve. \(r = 2(1 + \cos \theta)\)

49 Polar Arc Length · Level 3

Find the exact length of the curve. Use a graph to determine the parameter interval. \(r = \cos^4 \left(\dfrac{\theta}{4}\right)\)

50 Polar Arc Length · Level 3

Find the exact length of the curve. Use a graph to determine the parameter interval. \(r = \cos^2 \left(\dfrac{\theta}{2}\right)\)

51 Polar Arc Length - Numerical · Level 3

Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval. One loop of the curve \(r = \cos 2 \theta\)

52 Polar Arc Length - Numerical · Level 3

Use a calculator to find the length of the curve correct to four decimal places. \(r = \tan \theta\), \(\dfrac{\pi}{6} \leq \theta \leq \dfrac{\pi}{3}\)

53 Polar Arc Length - Numerical · Level 3

Use a calculator to find the length of the curve correct to four decimal places. \(r = \sin(6 \sin \theta)\)

54 Polar Arc Length - Numerical · Level 3

Use a calculator to find the length of the curve correct to four decimal places. \(r = \sin\left(\dfrac{\theta}{4}\right)\)

55 Polar Surface Area · Level 4

(a) Use Formula 10.2.6 to show that the area of the surface generated by rotating the polar curve \(r = f(\theta)\), \(a \leq \theta \leq b\) (where \(f'\) is continuous and \(0 \leq a < b \leq \pi\)) about the polar axis is \(S = \displaystyle\int_{a}^{b} 2 \pi r \sin \theta \sqrt{r^2 + \left(\dfrac{d r}{d \theta}\right)^2} d \theta\)

Answer for 55(a)

(b) Use the formula in part (a) to find the surface area generated by rotating the lemniscate \(r^2 = \cos 2 \theta\) about the polar axis.

Answer for 55(b)

Enter your answer directly below each part above.

56 Polar Surface Area · Level 4

(a) Find a formula for the area of the surface generated by rotating the polar curve \(r = f(\theta)\), \(a \leq \theta \leq b\) (where \(f'\) is continuous and \(0 \leq a < b \leq \pi\)), about the line \(\theta = \dfrac{\pi}{2}\).

Answer for 56(a)

(b) Find the surface area generated by rotating the lemniscate \(r^2 = \cos 2 \theta\) about the line \(\theta = \dfrac{\pi}{2}\).

Answer for 56(b)

Enter your answer directly below each part above.

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