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.