10: Polar Coordinates, Parametric Equations (Exercises)

These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

10.1: Polar Coordinates

10.1.1Plot these polar coordinate points on one graph: $(2,\pi/3)\), $(-3,\pi/2)\), $(-2,-\pi/4)\), $(1/2,\pi)\), $(1,4\pi/3)\), $(0,3\pi/2)$.

Find an equation in polar coordinates that has the same graph as the given equation in rectangular coordinates.

10.1.2 $y=3x$ (answer)

10.1.3 $y=-4$ (answer)

10.1.4 $xy^2=1$ (answer)

10.1.5 $x^2+y^2=5$ (answer)

10.1.6 $y=x^3$ (answer)

10.1.7 $y=\sin x$ (answer)

10.1.8 $y=5x+2$ (answer)

10.1.9 $x=2$ (answer)

10.1.10 $y=x^2+1$ (answer)

10.1.11 $y=3x^2-2x$ (answer)

10.1.12 $y=x^2+y^2$ (answer)

Sketch the curve.

10.1.13 \( r=\cos\theta$

10.1.14 \( r=\sin(\theta+\pi/4)$

10.1.15 \( r=-\sec\theta$

10.1.16 $r=\theta/2$, $\theta\ge0\)

10.1.17 $r=1+\theta^1/\pi^2$

10.1.18 $r=\cot\theta\csc\theta$

10.1.19 $r={1\over\sin\theta+\cos\theta}$

10.1.20 $r^2=-2\sec\theta\csc\theta$

Find an equation in rectangular coordinates that has the same graph as the given equation in polar coordinates.

10.1.21 $r=\sin(3\theta)$ (answer)

10.1.22 $r=\sin^2\theta$ (answer)

10.1.23 $r=\sec\theta\csc\theta$ (answer)

10.1.24 $r=\tan\theta$ (answer)

10.2: Slopes in polar coordinates

Compute $y'=dy/dx$ and $y''=d^2y/dx^2$

10.2.1 \(r=\theta\) (answer)

10.2.2 \(r=1+\sin\theta\) (answer)

10.2.3 \(r=\cos\theta\) (answer)

10.2.4 \(r=\sin\theta\) (answer)

10.2.5 \(r=\sec\theta\) (answer)

10.2.6 \(r=\sin(2\theta)\) (answer)

Sketch the curves over the interval $[0,2\pi]$ unless otherwise stated.

10.2.7 \(r=\sin\theta+\cos\theta\)

10.2.8 \(r=2+2\sin\theta\)

10.2.9 $r={3\over2}+\sin\theta$

10.2.10 \(r= 2+\cos\theta\)

10.2.11 $r={1\over2}+\cos\theta$

10.2.12 $r=\cos(\theta/2), 0\le\theta\le4\pi$

10.2.13 \(r=\sin(\theta/3), 0\le\theta\le6\pi\)

10.2.14 $r=\sin^2\theta$

10.2.15 $r=1+\cos^2(2\theta)$

10.2.16 $r=\sin^2(3\theta)$

10.2.17 $r=\tan\theta$

10.2.18 $r=\sec(\theta/2), 0\le\theta\le4\pi$

10.2.19 $r=1+\sec\theta$

10.2.20 $r={1\over 1-\cos\theta}$

10.2.21 $r={1\over 1+\sin\theta}$

10.2.22 $r=\cot(2\theta)$

10.2.23 $r=\pi/\theta, 0\le\theta\le\infty$

10.2.24 $r=1+\pi/\theta, 0\le\theta\le\infty$

10.2.25 $r=\sqrt{\pi/\theta}, 0\le\theta\le\infty$

10.3: Areas in polar coordinates

Find the area enclosed by the curve.

10.3.1 $r=\sqrt{\sin\theta}$ (answer)

10.3.2 $r=2+\cos\theta$ (answer)

10.3.3 $r=\sec\theta, \pi/6\le\theta\le\pi/3$ (answer)

10.3.4 $r=\cos\theta, 0\le\theta\le\pi/3$ (answer)

10.3.5 $r=2a\cos\theta, a>0$ (answer)

10.3.6 $r=4+3\sin\theta$ (answer)

10.3.7 Find the area inside the loop formed by \( r=\tan(\theta/2)$. (answer)

10.3.8 Find the area inside one loop of \( r=\cos(3\theta)$. (answer)

10.3.9 Find the area inside one loop of \( r=\sin^2\theta$. (answer)

10.3.10 Find the area inside the small loop of \( r=(1/2)+\cos\theta$. (answer)

10.3.11 Find the area inside $r=(1/2)+\cos\theta$, including the area inside the small loop. (answer)

10.3.12 Find the area inside one loop of \( r^2=\cos(2\theta)$. (answer)

10.3.13 Find the area enclosed by $r=\tan\theta$ and \( r={\csc\theta\over\sqrt2}$. (answer)

10.3.14 Find the area inside $r=2\cos\theta$ and outside $r=1$. (answer)

10.3.15 Find the area inside $r=2\sin\theta$ and above the line $r=(3/2)\csc\theta$. (answer)

10.3.16 Find the area inside $r=\theta\), $0\le\theta\le2\pi$. (answer)

10.3.17 Find the area inside $r=\sqrt{\theta}$, $0\le\theta\le2\pi$. (answer)

10.3.18 Find the area inside both \( r=\sqrt3\cos\theta$ and $r=\sin\theta$. (answer)

10.3.19 Find the area inside both $r=1-\cos\theta$ and $r=\cos\theta$. (answer)

10.3.20 The center of a circle of radius 1 is on the circumference of a circle of radius 2. Find the area of the region inside both circles. (answer)

10.3.21 Find the shaded area in figure 10.3.4. The curve is $r=\theta\), $0\le\theta\le3\pi$. (answer)

Figure 10.3.4. An area bounded by the spiral of Archimedes.

10.4: Parametric Equations

10.4.1 What curve is described by $x=t^2$, $y=t^4$? If $t$ is interpreted as time, describe how the object moves on the curve.

10.4.2 What curve is described by $x=3\cos t$, $y=3\sin t$? If $t$ is interpreted as time, describe how the object moves on the curve.

10.4.3 What curve is described by $x=3\cos t$, $y=2\sin t$? If $t$ is interpreted as time, describe how the object moves on the curve.

10.4.4 What curve is described by $x=3\sin t$, $y=3\cos t$? If $t$ is interpreted as time, describe how the object moves on the curve.

10.4.5 Sketch the curve described by $x=t^3-t$, $y=t^2$. If $t$ is interpreted as time, describe how the object moves on the curve.

10.4.6 A wheel of radius 1 rolls along a straight line, say the$x$-axis. A point$P$is located halfway between the center of the wheel and the rim; assume$P$starts at the point $(0,1/2)$. As the wheel rolls, $P$ traces a curve; find parametric equations for the curve.(answer)

10.4.7 A wheel of radius 1 rolls around the outside of a circle of radius 3. A point$P$on the rim of the wheel traces out a curve called a hypercycloid, as indicated in figure 10.4.3. Assuming$P$starts at the point $(3,0)$, find parametric equations for the curve. (answer)

Figure 10.4.3. A hypercycloid and a hypocycloid.

10.4.8 A wheel of radius 1 rolls around the inside of a circle of radius 3. A point$P$on the rim of the wheel traces out a curve called a hypocycloid, as indicated in figure 10.4.3. Assuming$P$starts at the point$(3,0)$, find parametric equations for the curve. (answer)

10.4.9 An involute of a circle is formed as follows: Imagine that a long (that is, infinite) string is wound tightly around a circle, and that you grasp the end of the string and begin to unwind it, keeping the string taut. The end of the string traces out the involute. Find parametric equations for this curve, using a circle of radius 1, and assuming that the string unwinds counter-clockwise and the end of the string is initially at$(1,0)$. Figure 10.4.4 shows part of the curve; the dotted lines represent the string at a few different times. (answer)

Figure 10.4.4. An involute of a circle.

10.5: Calculus with Parametric Equations

10.5.1 Consider the curve of exercise 6 in section 10.4. Find all values of $t$ for which the curve has a horizontal tangent line. (answer)

10.5.2 Consider the curve of exercise 6 in section 10.4. Find the area under one arch of the curve. (answer)

10.5.3 Consider the curve of exercise 6 in section 10.4. Set up an integral for the length of one arch of the curve. (answer)

10.5.4 Consider the hypercycloid of exercise 7 in section 10.4. Find all points at which the curve has a horizontal tangent line. (answer)

10.5.5 Consider the hypercycloid of exercise 7 in section 10.4. Find the area between the large circle and one arch of the curve. (answer)

10.5.6 Consider the hypercycloid of exercise 7 in section 10.4. Find the length of one arch of the curve. (answer)

10.5.7 Consider the hypocycloid of exercise 8 in section 10.4. Find the area inside the curve. (answer)

10.5.8 Consider the hypocycloid of exercise 8 in section 10.4. Find the length of one arch of the curve. (answer)

10.5.9 Recall the involute of a circle from exercise 9 in section 10.4. Find the point in the first quadrant in figure 10.4.4 at which the tangent line is vertical. (answer)

10.5.10 Recall the involute of a circle from exercise 9 in section 10.4. Instead of an infinite string, suppose we have a string of length $\pi$ attached to the unit circle at $(-1,0)\), and initially laid around the top of the circle with its end at $(1,0)$. If we grasp the end of the string and begin to unwind it, we get a piece of the involute, until the string is vertical. If we then keep the string taut and continue to rotate it counter-clockwise, the end traces out a semi-circle with center at $(-1,0)\), until the string is vertical again. Continuing, the end of the string traces out the mirror image of the initial portion of the curve; see figure 10.5.1. Find the area of the region inside this curve and outside the unit circle. (answer)

10.5.11 Find the length of the curve from the previous exercise, shown in figure 10.5.1. (answer)

10.5.12 Find the length of the spiral of Archimedes (figure 10.3.4) for $0\le\theta\le2\pi$. (answer)

Figure 10.5.1. A region formed by the end of a string.

Contributors

David Guichard (Whitman College)