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# 4.4E:

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## Area and Arc Length in Polar Coordinates

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

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

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

Solution: $$\displaystyle \frac{9}{2}∫^π_0sin^2θdθ$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Solution: $$\displaystyle 9π$$

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

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

Solution: $$\displaystyle \frac{9π}{4}$$

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

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

Solution: $$\displaystyle \frac{9π}{8}$$

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

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

Solution: $$\displaystyle \frac{18π−27\sqrt{3}}{2}$$

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

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

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

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

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

Solution: $$\displaystyle 32(4π−33√) 25) Inside \(\displaystyle r=1+cosθ$$ and outside $$\displaystyle r=cosθ$$

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

Solution: $$\displaystyle 2π−4$$

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 $$\displaystyle 0≤θ≤2π$$

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

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

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

Solution: $$\displaystyle \sqrt{2}∫^1_0e^θdθ$$

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 $$\displaystyle 0≤θ≤2$$

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

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

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

Solution: 32

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

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 $$\displaystyle 0≤θ≤\frac{π}{2}$$

Solution: 6.238

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 $$\displaystyle 0≤θ≤π$$

Solution: 2

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

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

Solution: 4.39

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 $$\displaystyle 0≤θ≤π$$

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

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

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 $$\displaystyle 0≤θ≤π$$

Solution: $$\displaystyle C=2π(\frac{3}{2})=3π$$ and $$\displaystyle ∫^π_03dθ=3π$$

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

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

Solution: $$\displaystyle C=2π(5)=10π$$ and $$\displaystyle ∫^π_010dθ=10π$$

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

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.

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

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

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

Solution: The slope is $$\displaystyle \frac{1}{\sqrt{3}}$$.

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

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

Solution: The slope is 0.

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

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

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

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

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

Solution: The slope is undefined at $$\displaystyle θ=\frac{π}{4}$$.

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 $$\displaystyle θ=\frac{π}{3}$$.

Solution: Slope = −1.

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

Soltuion: Slope is $$\displaystyle \frac{−2}{π}$$.

61) $$\displaystyle r=lnθ, θ=e$$

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

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) $$\displaystyle r^2=4cos(2θ)$$

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

65) $$\displaystyle r=2sin(2θ)$$

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

Solution: 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 $$\displaystyle (0,0)$$.

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