# 11.4E: Exercises for Section 11.4

• Gilbert Strang & Edwin “Jed” Herman
• OpenStax

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

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

2) Region enclosed by $$r=3\sin θ$$

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

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

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

$$\displaystyle\frac{3}{2}∫^{π/2}_0\sin^2(2θ)\,dθ$$

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

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

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

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

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

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

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

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

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

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

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

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 $$r=6\sin θ$$

$$9π\text{ units}^2$$

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

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

$$\frac{9π}{4}\text{ units}^2$$

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

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

$$\frac{9π}{8}\text{ units}^2$$

19) Enclosed by $$r=1+\sin θ$$

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

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

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

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

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

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

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

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

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

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

$$(2π−4)\text{ units}^2$$

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 $$0≤θ≤2π$$

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

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

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

$$\displaystyle\sqrt{2}∫^1_0e^θ\,dθ$$

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 $$0≤θ≤2$$

$$\frac{\sqrt{10}}{3}(e^6−1)$$ units

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

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

$$32$$ 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 $$0≤θ≤\frac{π}{2}$$

$$6.238$$ 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 $$0≤θ≤π$$

$$2$$ units

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

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

$$4.39$$ 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 $$0≤θ≤π$$

$$A=π\left(\frac{\sqrt{2}}{2}\right)^2=\dfrac{π}{2}\text{ units}^2$$ 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 $$0≤θ≤π$$

$$C=2π\left(\frac{3}{2}\right)=3π$$ 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 $$0≤θ≤π$$

$$C=2π(5)=10π$$ 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.

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

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

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

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

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

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

The slope is 0.

53) $$r=6+3\cos θ; \; (3,π)$$

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

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

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 $$θ=\frac{π}{3}$$.

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) $$r=θ, \; θ=\frac{π}{2}$$

Slope is $$\frac{−2}{π}$$.

61) $$r=\ln θ, \; θ=e$$

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

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) $$r^2=4\cos(2θ)$$

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 $$r=1+\sin θ$$

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.