Part 4: Parametric Equations and Polar Coordiantes

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4.1 Planar Parametrizations

Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
  • 10.1

4.1.1 Parametric Equations

• If \(x(t),y(t)\) are both defined as functions of \(t\) over some interval \(I\) of real numbers, then the set of points \(\{(x(t),y(t)):t\in I\}\) is the parametric curve with a system of parametric equations \(x(t),y(t)\). If \(I\) is omitted, then it is assumed that \(t\) belongs to the set of all real numbers.
• Example Plot the parametric curve \(x=\cos t,y=\sin t\) for \(0\leq t\leq 2\pi\), first by using a chart of \(t,x,y\) values, then by expressing the curve as an equation of \(x,y\).
• Example Show that the systems of parametric equations \(x_0=t,y_0=t^2\) and \(x_1=2t-2,y_1=4t^2-8t+4\) share the same parametric curve.

4.1.2 Parametrizing Curves Defined by Functions

• The curve \(y=f(x)\) where \(x\) belongs to the interval \(I\) may be easily parametrized left-to-right by the system of parametric equations \(x=t,y=f(t)\) where \(t\) also belongs to \(I\).
• Example Give a system of parametric equations for the curve \(y=\ln x\) from \((1,0)\) to \((e^2,2)\).

4.1.3 Parametrizing Line Segments

• The line segment joining the points \((x_0,y_0),(x_1,y_1)\) may be parametrized by \(x=x_0+(x_1-x_0)t,y=y_0+(y_1-y_0)t\) where \(0\leq t\leq 1\).
• Example Give a system of parametric equations for the line segment joining \((2,-3)\) and \((-1,4)\).
• Example Give two different systems of parametric equations for the portion of the line \(y=3x-2\) between \(x=-1\) and \(x=2\).
• The full line may be obtained with the same equations by allowing \(t\) to range over all real numbers.

Exercises for 4.1

1. Plot the parametric curve \(x=2-t^2,y=2t^2\) for \(0\leq t\leq 3\), first by using a chart of \(t,x,y\) values, then by expressing the curve as an equation of \(x,y\).
2. Plot the parametric curve \(x=3^t,y=3^{-t}\) for \(-\infty<t<\infty\), first by using a chart of \(t,x,y\) values, then by expressing the curve as an equation of \(x,y\).
3. Show that the systems of parametric equations \(x_0=t+2,y_0=e^2e^t\) and \(x_1=\ln t,y_1=t\) share the same parametric curve. Then plot that curve.
4. Give a system of parametric equations for the curve \(y=\cosh x\) from \((-\ln 2,5/4)\) to \((\ln 2,5/4)\).
5. Give a system of parametric equations for the line segment joining \((0,-4)\) and \((3,5)\).
6. Give a system of parametric equations for the line segment joining \((1,2)\) and \((-3,3)\).
7. Give two different systems of parametric equations for the portion of the line \(y=4-3x\) between \(x=-2\) and \(x=3\).
8. (Optional) Let \(a<b\). Find a system of parametric equations which parametrizes the planar curve \(y=f(x)\) right-to-left from \(x=b\) to \(x=a\).

Solutions

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4.2 Applications of Parametrizations

Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
  • 10.2, 6.3, 6.4

4.2.1 Parametric Formula for \(dy/dx\)

• Let \(y\) be a function of \(x\), and suppose its curve is parametrized by the equations \(x(t),y(t)\). Then by the Chain Rule, \(\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}\) at each point where all of these functions is differentiable.
• Example Find the line tangent to the curve parametrized by \(x=\tan t,y=\sec t,-\frac{\pi}{2}<t<\frac{\pi}{2}\) at the point \((1,\sqrt 2)\).
• Example Find the point on the parametric curve \(x=\ln t,y=t+\frac{1}{t},t>0\) which has a horizontal tangent line.

4.2.2 Arclength

• Suppose a curve \(C\) is defined parametrically by one-to-one functions \(x(t),y(t)\) on \(a\leq t\leq b\), where \(\frac{dx}{dt},\frac{dy}{dt}\) are continuous and never simultaneously zero. Then the length of \(C\) is defined to be \( L = \int_{t=a}^{t=b}ds = \int_{t=a}^{t=b}\sqrt{dx^2+dy^2} = \int_a^b\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\,dt \).
• Example Use the arclength formula to find the length of the line segment joining \((-2,3)\) and \((2,0)\).

• Example Find the perimeter of the curve parametrized by \(x=\sin^3 t,y=\cos^3 t,0\leq t\leq 2\pi\).

4.2.3 Surface Areas from Revolution

• Suppose a smooth curve \(C\) is defined parametrically by one-to-one functions \(x(t),y(t)\) on \(a\leq t\leq b\), with \(y(t)\geq 0\). Then the area of the surface of revolution obtained by rotating \(C\) about the \(x\)-axis is given by \(2\pi\int_{t=a}^{t=b}y(t)\,ds\).

• Example Find the area of the surface of revolution obtained by rotating the portion of the parabola \(y=x^2\) from \((0,0)\) to \((4,2)\) around the \(x\)-axis.

Exercises for 4.2

1. Find the line tangent to the curve parametrized by \(x=t^2,y=t^3\) at the point where \(t=-2\).
2. Show that the line tangent to the curve parametrized by \(x=3\sin t,y=3\cos t\) at the point \((\frac{3}{2},\frac{3\sqrt 3}{2})\) has the equation \(y=2\sqrt{3}-\frac{1}{\sqrt 3}x\). (Hint: \(\frac{1}{\sqrt 3}\frac{3}{2}=\frac{\sqrt 3}{2}\).)
3. Find the point on the parametric curve \(x=2t^2+1,y=t^4-4t\) which has a horizontal tangent line.
4. Use the arclength formula to find the length of the line segment joining \((-2,6)\) and \((3,-6)\).
5. Use the arclength formula to prove that the circumference of a circle of radius \(R\) is \(2\pi R\).
6. Show that the arclength of the curve parameterized by \(x=\cos 2t\), \(y=2t+\sin 2t\), \(0\leq t\leq \pi/2\) is \(4\). (Hint: \(1+\cos 2t=2\cos^2 t\).)
7. Find the area of the surface obtained by rotating the curve parameterized by \(x=\cos t,y=2+\sin t,0\leq t\leq \pi/2\) around the \(x\)-axis.
8. Use the parametric equations \(x=t,y=t,0\leq t\leq 1\) to show that the surface area of the cone of height \(1\) and radius \(1\) is \(\pi(\sqrt 2+1)\). (Hint: Don’t forget to add the area of the base of the cone.)
9. (Optional) Show that the surface area of the cone of height \(H\) and radius \(R\) is \(\pi R(\sqrt{H^2+R^2}+R)\).
10. (Quiz) Find the point on the parametric curve \(x=e^{3t}+5,y=e^{2t}-2t+1\) which has a horizontal tangent line.
    • \((e^3,e^2)\)
    • \((6,2)\)
    • \((5,e-1)\)
    • (Note: the quiz given in class had (6,0) as a choice by mistake, making the correct answer D.)
11. (Quiz) Which of these integrals gives the arclength of the curve \(y=x^2-3x+4\) between \((1,2)\) and \((3,4)\)?
    • \(\int_1^3\sqrt{4t^2-12t+10}\,dt\)
    • \(\int_2^4\sqrt{2t+3t^2}\,dt\)
    • \(\int_0^1(4+2t)\,dt\)

Solutions

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4.3 Polar Coordinates

Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
  • 10.3

4.3.1 Definition of Polar Coordinates

• The polar coordinate \(p(r,\theta)\) is defined to be Cartesian coordinate \((r\cos\theta,r\sin\theta)\).
• Example Plot the polar coordinates \(p(2,\pi/3),p(\sqrt{2},3\pi/4),p(-2,4\pi/3),p(4,11\pi/6)\).
• Note that every polar coordinate \(p(r,\theta)\) is equal to \(p(r,\theta+k\pi)\) for all even integers \(k\), and equal to \(p(-r,\theta+k\pi)\) for all odd integers \(k\).

4.3.2 Equations Relating Polar and Cartesian Coordinates

• Polar and Cartesian coordinates may be related by the equations \(x=r\cos\theta\), \(y=r\sin\theta\), \(x^2+y^2=r^2\), and \(\tan\theta=\frac{y}{x}\).
• Example Convert the Cartesian coordinate \((-2\sqrt3,2)\) into a polar coordinate.

• Example Convert the polar equation \(r=\frac{1}{\sin\theta-\cos\theta}\), \(\pi/4<\theta<5\pi/4\) into a Cartesian equation.
• Example Convert the Cartesian equation \((x-2)^2+y^2=4\) into a polar equation.

4.3.3 Common Polar Equations

• The equation \(r=R\) is a circle centered at the origin of radius \(R\).
• The equation \(\theta=\alpha\) is the line passing through the origin at the angle \(\alpha\).
• The equation \(r\cos\theta=a\) is the vertical line \(x=a\).
• The equation \(r\sin\theta=b\) is the horizontal line \(y=b\).
• Example Sketch the region where \(0<\csc\theta\leq r\leq 2\).
• The equations \(r=a\pm a\cos\theta\) and \(r=a\pm a\sin\theta\) are known as cardioids.
• Example Sketch the cardioid \(r=4-4\sin\theta\).

Exercises for 4.3

1. Convert the polar coordinates \(p(\sqrt 3,2\pi/3),p(\sqrt 2,\pi/4),p(2,7\pi/6),p(-\sqrt 3,-\pi/3)\) to Cartesian and plot them in the \(xy\) plane.
2. Convert the Cartesian coordinates \((4,-4),(-\frac{3}{2},-\frac{\sqrt 3}{2})\) into polar coordinates.
3. Convert the polar equation \(r=\frac{5}{\sqrt{25-9\sin^2\theta}}\) into a Cartesian equation. Name the curve.
4. Convert the Cartesian equation \(1-\frac{y}{x^2+y^2}=\frac{3}{\sqrt{x^2+y^2}}\) into a polar equation.
5. Convert the Cartesian equation for the line \(y=\frac{x}{\sqrt 3}\) into a polar equation.
6. Sketch the region where \(0< 3\sec\theta\leq r\leq 6\cos\theta\). (Hint: Completing the square in \(x^2-6x+y^2=0\) yields \((x-3)^2+y^2=9\).)
7. Sketch the cardioid \(r=3+3\sin\theta\).
8. Sketch the cardioids \(r=1+\cos\theta\) and \(r=1-\cos\theta\). At what points do they intersect?
9. (OPTIONAL) Sketch the “three-leaved rose” \(r=\sin 3\theta\).
10. (QUIZ) Which of these polar coordinates gives the point \((-\sqrt3,1)\)?
    • \(p(\sqrt2,3\pi/4)\)
    • \(p(\sqrt3,\pi/3)\)
    • \(p(2,5\pi/6)\)
11. (QUIZ) Convert the circle \(x^2+(y-4)^2=16\) into a polar equation.
    • \(r=16\)
    • \(r=8\sin\theta\)
    • \(r=12\cos^2\theta-4\sin^2\theta\)
12. (QUIZ) Which of these equations gives the curve drawn below?
    • \(r=3+3\cos\theta\)
    • \(r=3-3\sin\theta\)
    • \(r=3\tan\theta\)

Solutions 1-5

Solutions 6-12

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4.4 Areas and Lengths using Polar Coordinates

Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
  • 10.5

4.4.1 Area Between Polar Curves

• The area of the circle sector of angle \(\theta\) is given by \(A=\pi r^2\times\frac{\theta}{2\pi}=\frac{1}{2}r^2\theta\).
• Therefore the area bounded by \(\alpha\leq\theta\leq\beta\), \(r=f(\theta)\) is \(A=\frac{1}{2}\int_\alpha^\beta(f(\theta))^2\,d\theta\).
• Example Find the area bounded by the cardioid \(r=2+2\sin\theta\).

• To obtain the area where \(f(\theta)\leq r\leq g(\theta)\), where \(f\) is an inside curve and \(g\) is an outside curve, find the clockwise angle \(\alpha\) and counter-clockwise angle \(\beta\) where they intersect, and use \(A=\frac{1}{2}\int_\alpha^\beta((g(\theta))^2-(f(\theta))^2)\,d\theta\).
• Example Find the area outside the circle \(x^2+y^2=1\) and inside the cardioid \(r=1-\cos\theta\).

4.4.2 Length of a Polar Curve

• The polar curve \(r=f(\theta)\) where \(\alpha\leq\theta\leq\beta\) may be parametrized by \(x=f(\theta)\cos\theta,y=f(\theta)\sin\theta\) with the same bounds, using \(\theta\) in place of \(t\) as the parameter.
• Therefore its length is given by \(L=\int_\alpha^\beta\sqrt{ (\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2 }\,d\theta\) which simplifies to \(L=\int_\alpha^\beta\sqrt{ (f(\theta))^2+(f’(\theta))^2 }\,d\theta\).
• Example Show that the circumference of the circle of radius \(R\) is \(2\pi R\).
• Example Find the circumference of the spiral \(r=\theta^2\) from \(p(0,0)\) to \(p(5,\sqrt 5)\).

Exercises for 4.4

1. Find the area inside \(r=\cos2\theta\) where \(0\leq\theta\leq\pi/4\).
2. Find the area bounded by the cardioid \(r=1-\cos\theta\).
3. Sketch the region where \(|x|\leq y\leq\sqrt{1-x^2}+1\). Show that its area is \(\frac{\pi}{2}+1\). (Hint: Show that this is the area inside \(r=2\sin\theta\) where \(\pi/4\leq\theta\leq3\pi/4\).)
4. Find the length of one rotation of the spiral \(r=e^\theta\).
5. Use the polar arclength formula to show that the circumference of the circle \(r=4\sin\theta\) is \(4\pi\).
6. Show that the length of the cardioid \(r=2+2\cos\theta\) is \(\int_0^{2\pi}\sqrt{8+8\cos\theta}\,d\theta=16\).
7. (OPTIONAL) Prove that if \(x=f(\theta)\cos\theta\) and \(y=f(\theta)\sin\theta\) then \(\int_\alpha^\beta\sqrt{ (\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2 }\,d\theta =\int_\alpha^\beta\sqrt{ (f(\theta))^2+(f’(\theta))^2 }\,d\theta\).
8. (QUIZ) Which of these integrals is the area of the cardioid \(r=4+4\sin\theta\)?
   • \(\int_0^{2\pi}(8+16\sin\theta+8\sin^2\theta)\,d\theta\)
   • \(\int_0^{\pi/2}(16+16\sin^2\theta)\,d\theta\)
   • \(\int_0^{\pi}6\sin^2\theta\,d\theta\)
9. (QUIZ) Which of these integrals is the length of the curve \(r=\cos^2\theta\) where \(0\leq\theta\leq\pi/2\)?
   • \(\int_0^{\pi/2}4\sin\theta\sqrt{1-\cos^2\theta}\,d\theta\)
   • \(\int_0^{\pi/2}(\cos^4\theta-\pi)\,d\theta\)
   • \(\int_0^{\pi/2}\cos\theta\sqrt{1+3\sin^2\theta}\,d\theta\)

Solutions 1-7 Solutions 8-9