Stewart 9e Section 10.4: Calculus in Polar Coordinates

1 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Applications of Integration · Level 2

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

6 Applications of Integration · Level 2

Find the area of the shaded region shown in the figure.

7 Applications of Integration · Level 3

Find the area of the shaded region for \(r = \sqrt{\ln \theta}\), \(1 \leq \theta \leq 27\) (see figure).

8 Applications of Integration · Level 2

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

9 Applications of Integration · Level 2

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

10 Applications of Integration · Level 2

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

11 Applications of Integration · Level 2

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

12 Applications of Integration · Level 3

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

13 Applications of Integration · Level 3

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

14 Applications of Integration · Level 3

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

15 Applications of Integration · Level 4

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

16 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

17 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

18 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

19 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

20 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

21 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

22 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

23 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

24 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

25 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

26 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

27 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

28 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

29 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

30 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4

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

31 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5

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

32 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5

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

33 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5

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

34 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5

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

35 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5

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

36 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

37 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

38 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

39 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

40 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

41 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

42 Applications of Integration · Level 4

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.

43 Application - microphone pickup area · Level 4

When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. The microphone is placed 4 m from the front of the stage 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. Find the area on stage within the optimal pickup range of the microphone.

44 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2

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

45 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

46 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

47 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

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

48 Applications of Integration · Level 4

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

49 Applications of Integration · Level 4

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

50 Applications of Integration · Level 3

Use a calculator to find the length of one loop of the curve \(r = \cos(2 \theta)\) correct to four decimal places.

51 Applications of Integration · Level 3

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

52 Applications of Integration · Level 4

Use a calculator to find the length of the curve \(r = \sin(6 \sin \theta)\) correct to four decimal places. If necessary, graph the curve to determine the parameter interval.

53 Applications of Integration · Level 4

Use a calculator to find the length of the curve \(r = \sin\left(\dfrac{\theta}{4}\right)\) correct to four decimal places. If necessary, graph the curve to determine the parameter interval.

54 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5

(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 \(S = \displaystyle\int_{a}^{b} 2 \pi r \sin \theta \sqrt{r^2 + \left(\dfrac{d r}{d \theta}\right)^2} d \theta\). (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.

55 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5

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

56 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

Find the length of the cardioid \(r = 1 + \sin \theta\).

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