Part 4: Parametric Equations and Polar Coordiantes

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

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

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

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 $$0\leq\theta\leq\pi/2$$?
• $$\int_0^{\pi/2}4\sin\theta\sqrt{1-\cos^2\theta}\,d\theta$$
• $$\int_0^{\pi/2}(\cos^4\theta-\pi)\,d\theta$$
• $$\int_0^{\pi/2}\cos\theta\sqrt{1+3\sin^2\theta}\,d\theta$$