11.5E: Exercises for Section 11.5
- Page ID
- 160362
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.