Parametric Equations; Polar Equations

REVIEW EXERCISES
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In Problems 1–6:

(a) Find the rectangular equation of each curve.

(b) Graph each plane curve whose parametric equations are given and show its orientation.

(c) Determine the restrictions on $x$ and $y$ that make the rectangular equation identical to the plane curve.

$x(t) =4t-2, \, y(t) =1-t;\; -\infty <t<\infty$

$x=-4y+2$
(b)
(c) No restrictions

$x(t) =2t^{2}+6, \, y(t) =5-t;\;-\infty <t<\infty$

$x(t) =e^{t}, \, y(t) =e^{-t}\;-\infty < t< \infty$

$y=\dfrac{1}{x}$
(b)
$x\geq0$ , $y\geq0$

$x(t) =\ln t, \, y(t) =t^{3};\;t>0$

$x(t) =\sec ^{2}t, \, y(t) =\tan ^{2}t;\;0\leq t\leq \dfrac{\pi }{4}$

$y + 1 = x$
(b)
$1\leq x\leq2$ , $0\leq y\leq1$

$x(t) =t^{3/2}, \, y(t) =2t+4;\;t\geq 0$

In Problems 7–10, for the parametric equations below:

(a) Find an equation of the tangent line to the curve at $t$ .

(b) Graph the curve and the tangent line.

$x(t) =t^{2}-4,\; y(t) =t \,\, at \,\, t=1$

$y=\dfrac{1}{2}x+\dfrac{5}{2}$
(b)

$x(t) =3\sin t,\;y(t) =4\cos t+2 \,\, at \,\, t=\dfrac{\pi }{4}$

$x(t) =\dfrac{1}{t^{2}},\;y(t) =\sqrt{t^{2}+1} \,\, at \,\, t=3$

$y=-\dfrac{81}{2\sqrt{10}}x+\dfrac{9}{2\sqrt{10}}+\sqrt{10}$
(b)

$x(t) =\dfrac{t^{2}}{1+t},\;y(t) =\dfrac{t}{1+t} \,\, at \,\, t=0$

In Problems 11 and 12, find two different pairs of parametric equations for each rectangular equation.

$y=-2x+4$

Answers will vary.

$y=2x$

Describe the motion of an object that moves so that at time $t$ (in seconds) it has coordinates $x(t) =2\cos t,\;y(t) =\sin t, \;0\leq t\leq 2\pi$ .

$\dfrac{x^2}{4}+y=1$

In Problems 14–17, the polar coordinates of a point are given. Plot each point in a polar coordinate system, and find its rectangular coordinates.

$\left( 3,\dfrac{\pi }{6}\right)$

$\left( -2,\dfrac{4\pi }{3}\right)$

$\big(1, \sqrt{3}\big)$

$\left( 3,-\dfrac{\pi }{2}\right)$

$\left( -4,-\dfrac{\pi }{4}\right)$

$\big(-2\sqrt{2}, 2\sqrt{2}\big)$

In Problems 18–21, the rectangular coordinates of a point are given. Find two pairs of polar coordinates $( r,\theta )$ for each point, one with $r>0$ and the other with $r<0, 0 \le \theta < 2 \pi.$

$( 2,0)$

$( 3,4)$

$(5, 0.927)$ , $(-5, 4.068)$

$( -5,12)$

$( -3,3)$

$\left(3\sqrt{2}, \dfrac{3\pi}{4}\right), \left(-3\sqrt{2}, \dfrac{7\pi}{4}\right)$

In Problems 22–27, the letters $r, \theta$ represent polar coordinates. Write each equation in terms of the rectangular coordinates $x, y$ .

$r=4\sin ( 2\theta )$

$r=e^{\theta /2}$

$y=x \tan\ \ln(x^2+y^2)$

$r=\dfrac{1}{1+2\cos \theta }$

$r=a-\sin \theta$

$x^2 + y^2 = a \sqrt{x^2 + y^2} - y$

$r^{2}=4\cos ( 2\theta )$

$r=\theta$

$y=x\ \tan\sqrt{x^2+y^2}$

In Problems 28–31, the letters $x, y$ represent rectangular coordinates. Write each equation in terms of the polar coordinates $r, \theta$ .

$x^{2}+y^{2}=x$

$(x^{2}+y^{2})^{2}=x^{2}-y^{2}$

$\cos^2\theta - \sin^2\theta = r^2, r \neq 0$

$y^{2}=(x^{2}+y^{2})\cos ^{2}[(x^{2}+y^{2})^{1/2}]$

$\dfrac{x^{2}}{2^{2}}+\dfrac{y^{2}}{3^{2}}=1$

$r=\dfrac{6\sqrt{9\ \cos^2\theta+4\ \sin^2\theta}}{9\ \cos^2\theta+4\ \sin^2\theta}$

In Problems 32–35, identify and graph each polar equation. Convert it to a rectangular equation if necessary.

$r\sin \theta =1$

$r\sec \theta =2$

The circle centered at (1, 0) of radius 1.

$r=\sin \theta$

$r=-5\cos \theta$

The circle centered at $\left(-\dfrac{5}{2}, 0\right)$ of radius $\dfrac{5}{2}$ .

In Problems 36–45, for each equation:

(a) Graph the equation.

(b) Find parametric equations that represent the equation.

$r=1-\sin \theta$

$r=4\cos ( 2\theta )$

(a)
$x=4\ \cos(2\theta) \cos\theta\$ , $y=4\ \cos(2\theta) \sin\theta\$

$r=\dfrac{1}{2}-\sin \theta$

$r=\dfrac{4}{1-2\cos \theta }$

(a)
$x=\dfrac{4\ \cos\theta}{1-2\ \cos\theta}$ , $y=\dfrac{4\ \sin\theta}{1-2\ \cos\theta}$

$r=4\sin ( 3\theta )$

$r=2-2\cos \theta$

(a)
$x=(2-2\ \cos\theta)\cos\theta$ ,
$y=(2-2\ \cos\theta)\sin\theta\$

$r=2-\sin \theta$

$r=e^{0.5\theta }$

(a)
$x=e^{0.5\theta}\ \cos\theta$ ,
$y=e^{0.5\theta}\ \sin\theta$

$r^{2}=1-\sin ^{2}\theta$

$r^{2}=1+\sin ^{2}\theta$

(a)
$x=\sqrt{1+\sin^{2}\theta}\cos\theta$ ,
$y=\sqrt{1+\sin^{2}\theta}\sin\theta$

In Problems 46 and 47, for each polar equation:

(a) Identify and graph the equation.

(b) Convert the polar equation to a rectangular equation.

(c) Find parametric equations for the polar equation.

$r=\dfrac{2}{1-\cos \theta }$

$r=\dfrac{1}{1-\dfrac{1}{6}\cos \theta }$

(a) An ellipse
$35\left(x-\dfrac{6}{35}\right)^{2}+36y^{2}=\dfrac{1296}{35}$
$x=\dfrac{6\ \cos\theta}{6-\cos\theta}$ , $y=\dfrac{6\ \sin\theta}{6-\cos\theta}$

In Problems 48 and 49, find parametric equations for an object that moves along the ellipse $\dfrac{x^{2}}{16}+\dfrac{y^{2}}{9}=1$ with the motion described.

The motion begins at $( 4,0)$ , is counterclockwise, and requires 4 seconds for a complete revolution.

The motion begins at $( 0, 3)$ , is clockwise, and requires 5 seconds for a complete revolution.

$x=4\ \sin\ \left(\dfrac{2\pi t}{5}\right),\ y=3\ \cos\ \left(\dfrac{2\pi t}{5}\right),\ 0<t<5$

In Problems 50 and 51, find the points (if any) on the curve at which the tangent line is vertical or horizontal.

$x(t) =t^{3}-1,\;y(t) =2t^{2}+1$

$x(t) =1-\sin t,\;y(t) =2+3\cos t,\;0\leq t\leq 2\pi$

Vertical: $(0, 2)$ , $(2, 2)$ ; horizontal: $(1, 5)$ , $(1, -1)$

In Problems 52–55, find the arc length of each plane curve.

$x(t) =\sinh ^{-1}t, \, y(t) =\sqrt{t^{2}+1} \, {\rm from} \, t=0 \, {\rm to} \, t=1$

$x(t) =\tan t, \, y(t) =\dfrac{1}{3}( \sec ^{2}t+1) \, {\rm from} \, t=0 \, {\rm to} \, t=\dfrac{\pi }{4}$

$s=\dfrac{1}{3}\dfrac{\sqrt{13}}{2}+\dfrac{9}{8}\ln\left(1+\dfrac{\sqrt{13}}{2}\right)-\dfrac{9}{8}\ln\left(\dfrac{\sqrt{13}}{2}\right)$

$x(t) =e^{t}, \, y(t) =\dfrac{1}{2} e^{2t}-\dfrac{1}{4}\,t \, {\rm from} \, t=0 \, {\rm to} \, t=2$

$x=\dfrac{1}{2}\,y^{2}-\dfrac{1}{4}\ln y \, {\rm from} \, y=1\, {\rm to} \,y=2$

$s=\dfrac{3}{2}+\dfrac{1}{4}\ln2$

In Problems 56–59, find the arc length of each curve represented by a polar equation.

$r=2\sin \theta \, {\rm from} \, \theta =0\, {\rm to} \,\theta =\pi$

$r=e^{-\theta } \, {\rm from} \, \theta =0\, {\rm to} \,\theta =2\pi$

$\sqrt{2}(1-e^{-2\pi})$

$r=3\theta \, {\rm from} \, \theta =0\, {\rm to} \,\theta =2\pi$

$r=2\sin ^{2}\dfrac{\theta }{2} \, {\rm from} \, \theta =-\dfrac{\pi }{2}\, {\rm to} \,\theta =\dfrac{\pi }{2}$

$2\big(4-2\sqrt{2}\big)$

Area Find the area of the region inside the circle $r=4\sin \theta$ and above the line $r=3\csc \theta$ .

Area Find the area of the region that lies inside the rose $r=4\cos ( 2\theta )$ and outside the circle $r=\sqrt{2}$ .

$2\sqrt{7} + 12 \cos^{-1}\dfrac{\sqrt{2}}{4}$

Area Find the area of the region common to the graphs of $r=\cos \theta$ and $r=1-\cos \theta$ .

Surface Area Find the surface area of the solid generated by revolving the smooth curve represented by the parametric equations $x(t) =\sinh ^{-1}t,\;y(t) =\sqrt{t^{2}+1}$ from $0\leq \;t\leq 1$ about the $x$ -axis.

$\pi[\sqrt{2} + \ln(1+\sqrt{2})]$

Surface Area Find the surface area of the solid generated by revolving the smooth curve represented by $x(t) =e^{t}-t,\;y(t) =4e^{t/2}$ from $t=0$ to $t=1$ about the $x$ -axis.

Surface Area Find the surface area of the solid of revolution generated by revolving the curve represented by $x=y^{2}+1,\;0\leq y\leq 2$ , about the $y$ -axis.

Surface Area Find the surface area of the solid generated by revolving the curve represented by $y=\cos \, \left( \dfrac{x}{2}\right)\, ,\, 0\leq \, x\leq \pi$ about the $x$ -axis.

Surface Area Find the surface area of the solid of revolution generated by revolving the arc of the circle $r=4,\;0\leq \theta \leq \dfrac{\pi }{3}$ , about the $x$ -axis.

$16\pi$

REVIEW EXERCISESPrinted Page 9999

REVIEW EXERCISES
Printed Page 9999