11.5E: Exercises for Section 11.5

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

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

2) Region enclosed by \(r=3\sin θ\)

Answer

\(\displaystyle\frac{9}{2}∫^π_0\sin^2θ\,dθ\)

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

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

Answer

\(\displaystyle 32∫^{π/2}_0\sin^2(2θ)\,dθ\)

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

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

Answer

\(\displaystyle\frac{1}{2}∫^{2π}_π(1−\sin θ)^2\,dθ\)

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

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

Answer

\(\displaystyle∫^{π/2}_{\sin^{−1}(2/3)}(2−3\sin θ)^2\,dθ\)

9) Region enclosed by the inner loop of \(r=3−4\cos θ\)

10) Region enclosed by \(r=1−2\cos θ\) and outside the inner loop

Answer

\(\displaystyle∫^π_0(1−2\cos θ)^2\,dθ−∫^{π/3}_0(1−2\cos θ)^2\,dθ\)

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

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

Answer

\(\displaystyle4∫^{π/3}_0\,dθ+16∫^{π/2}_{π/3}(\cos^2θ)\,dθ\)

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

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

14) Enclosed by \(r=6\sin θ\)

Answer

\(9π\text{ units}^2\)

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

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

Answer

\(\frac{9π}{4}\text{ units}^2\)

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

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

Answer

\(\frac{9π}{8}\text{ units}^2\)

19) Enclosed by \(r=1+\sin θ\)

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

Answer

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

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

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

Answer

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

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

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

Answer

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

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

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

Answer

\((2π−4)\text{ units}^2\)

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

27) \(r=4\cos θ\) on the interval \(0≤θ≤\frac{π}{2}\)

28) \(r=1+\sin θ\) on the interval \(0≤θ≤2π\)

Answer

\(\displaystyle∫^{2π}_0\sqrt{(1+\sin θ)^2+\cos^2θ}\,dθ\)

29) \(r=2\sec θ\) on the interval \(0≤θ≤\frac{π}{3}\)

30) \(r=e^θ\) on the interval \(0≤θ≤1\)

Answer

\(\displaystyle\sqrt{2}∫^1_0e^θ\,dθ\)

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

31) \(r=6\) on the interval \(0≤θ≤\frac{π}{2}\)

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

Answer

\(\frac{\sqrt{10}}{3}(e^6−1)\) units

33) \(r=6\cos θ\) on the interval \(0≤θ≤\frac{π}{2}\)

34) \(r=8+8\cos θ\) on the interval \(0≤θ≤π\)

Answer

\(32\) units

35) \(r=1−\sin θ\) on the interval \(0≤θ≤2π\)

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

36) [T] \(r=3θ\) on the interval \(0≤θ≤\frac{π}{2}\)

Answer

\(6.238\) units

37) [T] \(r=\dfrac{2}{θ}\) on the interval \(π≤θ≤2π\)

38) [T] \(r=\sin^2\left(\frac{θ}{2}\right)\) on the interval \(0≤θ≤π\)

Answer

\(2\) units

39) [T] \(r=2θ^2\) on the interval \(0≤θ≤π\)

40) [T] \(r=\sin(3\cos θ)\) on the interval \(0≤θ≤π\)

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) \(r=3\sin θ\) on the interval \(0≤θ≤π\)

42) \(r=\sin θ+\cos θ\) on the interval \(0≤θ≤π\)

Answer

\(A=π\left(\frac{\sqrt{2}}{2}\right)^2=\dfrac{π}{2}\text{ units}^2\) and \(\displaystyle\frac{1}{2}∫^π_0(1+2\sin θ\cos θ)\,dθ=\frac{π}{2}\text{ units}^2\)

43) \(r=6\sin θ+8\cos θ\) on the interval \(0≤θ≤π\)

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) \(r=3\sin θ\) on the interval \(0≤θ≤π\)

Answer

\(C=2π\left(\frac{3}{2}\right)=3π\) units and \(\displaystyle∫^π_03\,dθ=3π\) units

45) \(r=\sin θ+\cos θ\) on the interval \(0≤θ≤π\)

46) \(r=6\sin θ+8\cos θ\) on the interval \(0≤θ≤π\)

Answer

\(C=2π(5)=10π\) units and \(\displaystyle∫^π_010\,dθ=10π\) units

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

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

48) Use the definition of the derivative \(\dfrac{dy}{dx}=\dfrac{dy/dθ}{dx/dθ}\) and the product rule to derive the derivative of a polar equation.

Answer

\(\dfrac{dy}{dx}=\dfrac{f′(θ)\sin θ+f(θ)\cos θ}{f′(θ)\cos θ−f(θ)\sin θ}\)

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

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

Answer

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

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

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

Answer

The slope is 0.

53) \(r=6+3\cos θ; \; (3,π)\)

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

Answer

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

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

56) \(r=2θ; \; \left(\frac{π}{2},\frac{π}{4}\right)\)

Answer

The slope is undefined at \(θ=\frac{π}{4}\).

57) Find the points on the interval \(−π≤θ≤π\) at which the cardioid \(r=1−\cos θ\) has a vertical or horizontal tangent line.

58) For the cardioid \(r=1+\sin θ,\) find the slope of the tangent line when \(θ=\frac{π}{3}\).

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 \(θ\).

59) \(r=3\cos θ,\; θ=\frac{π}{3}\)

60) \(r=θ, \; θ=\frac{π}{2}\)

Answer

Slope is \(\frac{−2}{π}\).

61) \(r=\ln θ, \; θ=e\)

62) [T] Use technology: \(r=2+4\cos θ\) at \(θ=\frac{π}{6}\)

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) \(r=4\cos θ\)

64) \(r^2=4\cos(2θ)\)

Answer

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

65) \(r=2\sin(2θ)\)

66) The cardioid \(r=1+\sin θ\)

Answer

Horizontal tangents at \(\frac{π}{2},\, \frac{7π}{6},\, \frac{11π}{6}.\)
Vertical tangents at \(\frac{π}{6},\, \frac{5π}{6}\) and also at the pole \((0,0)\).

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