6.4E: Excercises

Area and Arc Length in Polar Coordinates

Answered to even-numbered problems provided.

Exercise \(\PageIndex{1}\)

For the following exercises, determine a definite integral that represents the area.

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

2) Region enclosed by \(\displaystyle r=3sinθ\)

3) Region in the first quadrant within the cardioid \(\displaystyle r=1+sinθ\)

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

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

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

7) Region in the first quadrant enclosed by \(\displaystyle r=2−cosθ\)

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

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

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

11) Region common to \(\displaystyle r=3sinθ\) and \(\displaystyle r=2−sinθ\)

12) Region common to \(\displaystyle r=2\) and \(\displaystyle r=4cosθ\)

13) Region common to \(\displaystyle r=3cosθ\) and \(\displaystyle r=3sinθ\)

Answer

Solution 2: \(\displaystyle \frac{9}{2}∫^π_0sin^2θdθ\),

Solution 4: \(\displaystyle \frac{3}{2}∫^{π/2}_0sin^2(2θ)dθ\),

Solution 6: \(\displaystyle \frac{1}{2}∫^{2π}_π(1−sinθ)^2dθ\),

Solution 8: \(\displaystyle ∫^{π/2}_{sin^{−1}(2/3)}(2−3sinθ)^2dθ\),

Solution 10: \(\displaystyle ∫^π_0(1−2cosθ)^2dθ−∫^{π/3}_0(1−2cosθ)^2dθ\),

Solution 12: \(\displaystyle 4∫^{π/3}_0dθ+16∫^{π/2}_{π/3}(cos^2θ)dθ\)

Exercise \(\PageIndex{2}\)

For the following exercises, find the area of the described region.

14) Enclosed by \(\displaystyle r=6sinθ\)

15) Above the polar axis enclosed by \(\displaystyle r=2+sinθ\)

16) Below the polar axis and enclosed by \(\displaystyle r=2−cosθ\)

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

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

19) Enclosed by \(\displaystyle r=1+sinθ\)

20) Enclosed by the inner loop of \(\displaystyle r=3+6cosθ\)

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

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

23) Common interior of \(\displaystyle r=3−2sinθ\) and \(\displaystyle r=−3+2sinθ\)

24) Common interior of \(\displaystyle r=6sinθ\) and \(\displaystyle r=3\)

25) Inside \(\displaystyle r=1+cosθ\) and outside \(\displaystyle r=cosθ\)

26) Common interior of \(\displaystyle r=2+2cosθ\) and \(\displaystyle r=2sinθ\)

Answer

Solution 14: \(\displaystyle 9π\),

Solution 16: \(\displaystyle \frac{9π}{4}\),

Solution 18: \(\displaystyle \frac{9π}{8}\),

Solution 20: \(\displaystyle \frac{18π−27\sqrt{3}}{2}\),

Solution 22: \(\displaystyle \frac{4}{3}(4π−3\sqrt{3})\),

Solution 24: \(\displaystyle 32(4π−33√),

Solution 26: \(\displaystyle 2π−4\)

Exercise \(\PageIndex{3}\)

For the following exercises, find a definite integral that represents the arc length.

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

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

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

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

Answer

Solution 28: \(\displaystyle ∫^{2π}_0\sqrt{(1+sinθ)^2+cos^2θ}dθ\),

Solution 30: \(\displaystyle \sqrt{2}∫^1_0e^θdθ\)

Exercise \(\PageIndex{4}\)

For the following exercises, find the length of the curve over the given interval.

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

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

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

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

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

Answer

Solution 32: \(\displaystyle \frac{\sqrt{10}}{3}(e^6−1)\),

Solution 34: 32

Exercise \(\PageIndex{5}\)

For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.

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

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

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

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

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

Answer

Solution 36: 6.238, Solution 38: 2, Solution 40: 4.39

Exercise \(\PageIndex{6}\)

For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.

41) \(\displaystyle r=3sinθ\) on the interval \(\displaystyle 0≤θ≤π\)

42) \(\displaystyle r=sinθ+cosθ\) on the interval \(\displaystyle 0≤θ≤π\)

43) \(\displaystyle r=6sinθ+8cosθ\) on the interval \(\displaystyle 0≤θ≤π\)

Answer

Solution 42: \(\displaystyle A=π(\frac{\sqrt{2}}{2})^2=\frac{π}{2}\) and \(\displaystyle \frac{1}{2}∫^π_0(1+2sinθcosθ)dθ=\frac{π}{2}\)

Exercise \(\PageIndex{7}\)

For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.

44) \(\displaystyle r=3sinθ\) on the interval \(\displaystyle 0≤θ≤π\)

45) \(\displaystyle r=sinθ+cosθ\) on the interval \(\displaystyle 0≤θ≤π\)

46) \(\displaystyle r=6sinθ+8cosθ\) on the interval \(\displaystyle 0≤θ≤π\)

47) Verify that if \(\displaystyle y=rsinθ=f(θ)sinθ\) then \(\displaystyle \frac{dy}{dθ}=f'(θ)sinθ+f(θ)cosθ.\)

Answer

Solution 44: \(\displaystyle C=2π(\frac{3}{2})=3π\) and \(\displaystyle ∫^π_03dθ=3π\), Solution 46: \(\displaystyle C=2π(5)=10π\) and \(\displaystyle ∫^π_010dθ=10π\)

Exercise \(\PageIndex{8}\)

For the following exercises, find the slope of a tangent line to a polar curve \(\displaystyle r=f(θ)\). Let \(\displaystyle x=rcosθ=f(θ)cosθ\) and \(\displaystyle y=rsinθ=f(θ)sinθ\), so the polar equation \(\displaystyle r=f(θ)\) is now written in parametric form.

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

49) \(\displaystyle r=1−sinθ; (\frac{1}{2},\frac{π}{6})\)

50) \(\displaystyle r=4cosθ; (2,\frac{π}{3})\)

51) \(\displaystyle r=8sinθ; (4,\frac{5π}{6})\)

52) \(\displaystyle r=4+sinθ; (3,\frac{3π}{2})\)

53) \(\displaystyle r=6+3cosθ; (3,π)\)

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

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

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

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

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

Answer

Solution 48: \(\displaystyle \frac{dy}{dx}=\frac{f′(θ)sinθ+f(θ)cosθ}{f′(θ)cosθ−f(θ)sinθ}\),

Solution 50: The slope is \(\displaystyle \frac{1}{\sqrt{3}}\),

Solution 52: The slope is 0,

Solution 54: At \(\displaystyle (4,0),\) the slope is undefined. At \(\displaystyle (−4,\frac{π}{2})\), the slope is 0,

Solution 56: The slope is undefined at \(\displaystyle θ=\frac{π}{4}\),

Solution: Slope = −1.

Exercise \(\PageIndex{9}\)

For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of \(\displaystyle θ\).

59) \(\displaystyle r=3cosθ,θ=\frac{π}{3}\)

60) \(\displaystyle r=θ, θ=\frac{π}{2}\)

61) \(\displaystyle r=lnθ, θ=e\)

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

Solution: Calculator answer: −0.836.

Answer

Soltuion 60: Slope is \(\displaystyle \frac{−2}{π}\), Solution 62: Calculator answer: −0.836.

Exercise \(\PageIndex{10}\)

For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line.

63) \(\displaystyle r=4cosθ\)

64) \(\displaystyle r^2=4cos(2θ)\)

65) \(\displaystyle r=2sin(2θ)\)

66) The cardioid \(\displaystyle r=1+sinθ\)

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

Answer

Solution 64: Horizontal tangent at \(\displaystyle (±\sqrt{2},\frac{π}{6}), (±\sqrt{2},−\frac{π}{6})\),

Solution 66: Horizontal tangents at \(\displaystyle \frac{π}{2},\frac{7π}{6},\frac{11π}{6}.\) Vertical tangents at \(\displaystyle \frac{π}{6},\frac{5π}{6}\) and also at the pole \(\displaystyle (0,0)\).