Chapter 10 ยท Introduction

Chapter 10 โ€” Parametric Equations and Polar Coordinates

When a point moves, time is part of the story โ€” curves traced by $x=f(t)$, $y=g(t)$, and curves drawn by turning and stretching with $r=f(\theta)$.

Throw a ball, ride a bicycle, or watch a planet drift across the night sky over a few months. In each case something moves, and the most natural way to describe it is to say where it is at each moment: its horizontal position is one function of time and its vertical position is another. A football's path, the looping track of the valve on a bicycle wheel, the oval orbit of a planet around the Sun โ€” all of these are pairs of functions $x=f(t)$, $y=g(t)$ sharing a common parameter $t$. You met this idea briefly in Calculus I, Section 2.5, where we found slopes of such curves. This chapter takes it much further.

A graph $y=f(x)$ is a surprisingly limited tool. It cannot loop back on itself, it cannot go straight up, it cannot pass the same $x$-value twice, and it forgets when and how fast the point got there. Parametric equations remove every one of these restrictions. And for shapes that are naturally described by turning โ€” a radar sweep, a spiral, the petals of a flower, the beam pattern of an antenna โ€” polar coordinates $(r,\theta)$ are even more convenient. We will learn to draw these curves, find their tangent lines, and compute the areas they enclose and the lengths of their arcs.

Curves from a parameterโ†’Eliminating the parameterโ†’Slopes & tangentsโ†’Area & arc lengthโ†’Polar coordinatesโ†’Polar area & length
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • sketch a parametric curve, show its orientation, and convert between parametric and Cartesian equations;
  • write parametric equations for lines, circles, ellipses, cycloids and projectile paths;
  • find $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$ for a parametric curve and locate its horizontal and vertical tangents;
  • compute areas, arc lengths, speeds and surface areas of revolution for parametric curves;
  • plot points and curves in polar coordinates, convert between polar and rectangular forms, and find slopes and tangent lines of polar curves;
  • find areas enclosed by and between polar curves and the lengths of polar arcs.
Teaser โ€” The Path Remembers the Motion

Each motion below is described by a pair of functions $x(t)$, $y(t)$. Watch the point trace its path: the arrow shows its velocity, and its length shows the speed. Notice which of these paths could never be written as a single graph $y=f(x)$.

time $t$โ€”
position $(x,y)$โ€”
speedโ€”
a graph $y=f(x)$?โ€”

Sections in this chapter

10.1

Parametric Equations

Curves traced by a moving point; orientation; eliminating the parameter; cycloids, projectiles and Lissajous figures.

$x=r(\theta-\sin\theta),\ y=r(1-\cos\theta)$
Open section โ†’10.2

Calculus of Parametric Curves

Tangents and concavity, area under a parametric curve, arc length, speed and surface area.

$L=\int_a^b\sqrt{x^\prime(t)^2+y^\prime(t)^2}\,dt$
Open section โ†’10.3

Polar Coordinates

Locating points by distance and angle; converting to and from rectangular form; roses, cardioids and spirals; slopes and tangents of polar curves.

$x=r\cos\theta,\ y=r\sin\theta$
Open section โ†’10.4

Area and Arc Length in Polar Coordinates

Sweeping out area with thin sectors, areas between two polar curves, and measuring the length of a polar curve.

$A=\tfrac12\int_\alpha^\beta r^2\,d\theta$
Open section โ†’
โœ… Before you start
You will use the identity $\sin^2t+\cos^2t=1$ and the double-angle formulas constantly (Calculus I, 0.1), the chain rule (2.3) and parametric slopes $\dfrac{dy}{dx}=\dfrac{y^\prime(t)}{x^\prime(t)}$ (2.5). For the integrals, review trigonometric integrals (5.1), areas between curves (6.1) and arc length and surface area for graphs $y=f(x)$ (6.4) โ€” the parametric formulas in 10.2 generalise them.
โ† 9.4 Working with Taylor Series10.1 Parametric Equations โ†’