5.4E: Exercises

In exercises 1 -13, determine a definite integral that represents the area.

1) Region enclosed by \(r=4\)

2) Region enclosed by \(r=3\sin \theta \)

Answer

\(\displaystyle\frac{9}{2} \int ^ \pi _0\sin^2 \theta \,d \theta \)

3) Region in the first quadrant within the cardioid \(r=1+\sin \theta \)

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

Answer

\(\displaystyle\frac{3}{2} \int ^{ \pi /2}_0\sin^2(2 \theta )\,d \theta \)

5) Region enclosed by one petal of \(r=cos(3 \theta )\)

6) Region below the polar axis and enclosed by \(r=1−\sin \theta \)

Answer

\(\displaystyle\frac{1}{2} \int ^{2 \pi }_ \pi (1−\sin \theta )^2\,d \theta \)

7) Region in the first quadrant enclosed by \(r=2−\cos \theta \)

8) Region enclosed by the inner loop of \(r=2−3\sin \theta \)

Answer

\(\displaystyle \int ^{ \pi /2}_{\sin^{−1}(2/3)}(2−3\sin \theta )^2\,d \theta \)

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

Answer

\(\displaystyle \int ^ \pi _0(1−2\cos \theta )^2\,d \theta − \int ^{ \pi /3}_0(1−2\cos \theta )^2\,d \theta \)

11) Region common to \(r=3\sin \theta \) and \(r=2−\sin \theta \)

12) Region common to \(r=2\) and \(r=4\cos \theta \)

Answer

\(\displaystyle4 \int ^{ \pi /3}_0\,d \theta +16 \int ^{ \pi /2}_{ \pi /3}(\cos^2 \theta )\,d \theta \)

13) Region common to \(r=3\cos \theta \) and \(r=3\sin \theta \)

In exercises 14 -26, find the area of the described region.

14) Enclosed by \(r=6\sin \theta \)

Answer

\(9 \pi \text{ units}^2\)

15) Above the polar axis enclosed by \(r=2+\sin \theta \)

16) Below the polar axis and enclosed by \(r=2−\cos \theta \)

Answer

\(\frac{9 \pi }{4}\text{ units}^2\)

17) Enclosed by one petal of \(r=4\cos(3 \theta )\)

18) Enclosed by one petal of \(r=3\cos(2 \theta )\)

Answer

\(\frac{9 \pi }{8}\text{ units}^2\)

19) Enclosed by \(r=1+\sin \theta \)

20) Enclosed by the inner loop of \(r=3+6\cos \theta \)

Answer

\(\frac{18 \pi −27\sqrt{3}}{2}\text{ units}^2\)

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

22) Common interior of \(r=4\sin(2 \theta )\) and \(r=2\)

Answer

\(\frac{4}{3}(4 \pi −3\sqrt{3})\text{ units}^2\)

23) Common interior of \(r=3−2\sin \theta \) and \(r=−3+2\sin \theta \)

24) Common interior of \(r=6\sin \theta \) and \(r=3\)

Answer

\(\frac{3}{2}(4 \pi −3\sqrt{3})\text{ units}^2\)

25) Inside \(r=1+\cos \theta \) and outside \(r=\cos \theta \)

26) Common interior of \(r=2+2\cos \theta \) and \(r=2\sin \theta \)

Answer

\((2 \pi −4)\text{ units}^2\)

In exercises 27 - 30, find a definite integral that represents the arc length.

27) \(r=4\cos \theta \) on the interval \(0 \leq \theta \leq \frac{ \pi }{2}\)

28) \(r=1+\sin \theta \) on the interval \(0 \leq \theta \leq 2 \pi \)

Answer

\(\displaystyle \int ^{2 \pi }_0\sqrt{(1+\sin \theta )^2+\cos^2 \theta }\,d \theta \)

29) \(r=2\sec \theta \) on the interval \(0 \leq \theta \leq \frac{ \pi }{3}\)

30) \(r=e^ \theta \) on the interval \(0 \leq \theta \leq 1\)

Answer

\(\displaystyle\sqrt{2} \int ^1_0e^ \theta \,d \theta \)

In exercises 31 - 35, find the length of the curve over the given interval.

31) \(r=6\) on the interval \(0 \leq \theta \leq \frac{ \pi }{2}\)

32) \(r=e^{3 \theta }\) on the interval \(0 \leq \theta \leq 2\)

Answer

\(\frac{\sqrt{10}}{3}(e^6−1)\) units

33) \(r=6\cos \theta \) on the interval \(0 \leq \theta \leq \frac{ \pi }{2}\)

34) \(r=8+8\cos \theta \) on the interval \(0 \leq \theta \leq \pi \)

Answer

\(32\) units

35) \(r=1−\sin \theta \) on the interval \(0 \leq \theta \leq 2 \pi \)

In exercises 36 - 40, use the integration capabilities of a calculator to approximate the length of the curve.

36) [Technology Required] \(r=3 \theta \) on the interval \(0 \leq \theta \leq \frac{ \pi }{2}\)

Answer

\(6.238\) units

37) [Technology Required] \(r=\dfrac{2}{ \theta }\) on the interval \( \pi \leq \theta \leq 2 \pi \)

38) [Technology Required] \(r=\sin^2\left(\frac{ \theta }{2}\right)\) on the interval \(0 \leq \theta \leq \pi \)

Answer

\(2\) units

39) [Technology Required] \(r=2 \theta ^2\) on the interval \(0 \leq \theta \leq \pi \)

40) [Technology Required] \(r=\sin(3\cos \theta )\) on the interval \(0 \leq \theta \leq \pi \)

Answer

\(4.39\) units

In exercises 41 - 43, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.

41) \(r=3\sin \theta \) on the interval \(0 \leq \theta \leq \pi \)

42) \(r=\sin \theta +\cos \theta \) on the interval \(0 \leq \theta \leq \pi \)

Answer

\(A= \pi \left(\frac{\sqrt{2}}{2}\right)^2=\dfrac{ \pi }{2}\text{ units}^2\) and \(\displaystyle\frac{1}{2} \int ^ \pi _0(1+2\sin \theta \cos \theta )\,d \theta =\frac{ \pi }{2}\text{ units}^2\)

43) \(r=6\sin \theta +8\cos \theta \) on the interval \(0 \leq \theta \leq \pi \)

In exercises 44 - 46, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.

44) \(r=3\sin \theta \) on the interval \(0 \leq \theta \leq \pi \)

Answer

\(C=2 \pi \left(\frac{3}{2}\right)=3 \pi \) units and \(\displaystyle \int ^ \pi _03\,d \theta =3 \pi \) units

45) \(r=\sin \theta +\cos \theta \) on the interval \(0 \leq \theta \leq \pi \)

46) \(r=6\sin \theta +8\cos \theta \) on the interval \(0 \leq \theta \leq \pi \)

Answer

\(C=2 \pi (5)=10 \pi \) units and \(\displaystyle \int ^ \pi _010\,d \theta =10 \pi \) units

47) Verify that if \(y=r\sin \theta =f( \theta )\sin \theta \) then \(\dfrac{dy}{d \theta }=f'( \theta )\sin \theta +f( \theta )\cos \theta .\)

In exercises 48 - 56, find the slope of a tangent line to a polar curve \(r=f( \theta )\). Let \(x=r\cos \theta =f( \theta )\cos \theta \) and \(y=r\sin \theta =f( \theta )\sin \theta \), so the polar equation \(r=f( \theta )\) is now written in parametric form.

48) Use the definition of the derivative \(\dfrac{dy}{dx}=\dfrac{dy/d \theta }{dx/d \theta }\) and the Product Rule to derive the derivative of a polar equation.

Answer

\(\dfrac{dy}{dx}=\dfrac{f^{\prime}( \theta )\sin \theta +f( \theta )\cos \theta }{f^{\prime}( \theta )\cos \theta −f( \theta )\sin \theta }\)

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

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

Answer

The slope is \(\frac{1}{\sqrt{3}}\).

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

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

Answer

The slope is 0.

53) \(r=6+3\cos \theta ; \; (3, \pi )\)

54) \(r=4\cos(2 \theta );\) tips of the leaves

Answer

At \((4,0),\) the slope is undefined. At \(\left(−4,\frac{ \pi }{2}\right)\), the slope is 0.

55) \(r=2\sin(3 \theta );\) tips of the leaves

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

Answer

The slope is undefined at \( \theta =\frac{ \pi }{4}\).

57) Find the points on the interval \(− \pi \leq \theta \leq \pi \) 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 \( \theta =\frac{ \pi }{3}\).

Answer

Slope = −1.

In exercises 59 - 62, 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) \(r= \theta , \; \theta =\frac{ \pi }{2}\)

Answer

Slope is \(\frac{−2}{ \pi }\).

61) \(r=\ln \theta , \; \theta =e\)

62) [Technology Required] Use technology: \(r=2+4\cos \theta \) at \( \theta =\frac{ \pi }{6}\)

Answer

Calculator answer: −0.836.

In exercises 63 - 66, find the points at which the following polar curves have a horizontal or vertical tangent line.

63) \(r=4\cos \theta \)

64) \(r^2=4\cos(2 \theta )\)

Answer

Horizontal tangent at \(\left( \pm \sqrt{2},\frac{ \pi }{6}\right), \; \left( \pm \sqrt{2},−\frac{ \pi }{6}\right)\).

65) \(r=2\sin(2 \theta )\)

66) The cardioid \(r=1+\sin \theta \)

Answer

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 \((0,0)\).

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