E078 Section 8 — Parametric Equations and Polar Coordinates

Calculus 2 for Engineers (WGU E078), Section 8: parametric curves and calculus, polar coordinates, area and arc length in polar coordinates. Built from the zyBooks Section 8 study materials; not official WGU material.

1

Parametric equations (definition)

2

Orientation of a parametric curve

3

Eliminating the parameter

4

Eliminate $t$: $x = t^2 - 3$, $y = 2t + 1$

5

Eliminate $t$: $x = 3\cos t$, $y = 4\sin t$

6

Eliminate $t$: $x = \sec t$, $y = \tan t$

7

Eliminate $t$: $x = \sqrt{t}$, $y = 1 - t$

8

How many parameterizations does a curve have?

9

Two parameterizations of $y = x^2 + 2x$

10

Parametric circle, center $(h, k)$, radius $r$

11

Cycloid (wheel of radius $a$)

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Hypocycloid

13

Skill Shop: segment from $(2, 3)$ at $t = 0$ to $(5, 9)$ at $t = 2$

14

Skill Shop: part of $x^2 + y^2 = 4$ in Quadrant III

15

Skill Shop: circle center $(-3, 2)$, radius $5$

16

Derivative of parametric equations

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Horizontal tangent on a parametric curve

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Vertical tangent on a parametric curve

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Critical point of a parametric curve

20

$\frac{dy}{dx}$ for $x = 5\cos t$, $y = 5\sin t$

21

Second derivative of parametric equations

22

Is $\frac{d^2y}{dx^2} = \frac{y''(t)}{x''(t)}$?

23

$\frac{d^2y}{dx^2}$ for $x = t^2 - 4t$, $y = 2t^3 - 6t$

24

Area under a parametric curve

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Area under one arch of $x = t - \sin t$, $y = 1 - \cos t$

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Arc length of a parametric curve

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Arc length of $x = 3\cos t$, $y = 3\sin t$, $0 \le t \le \pi$

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Surface area: parametric curve revolved about the $x$-axis

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Surface area: parametric curve revolved about the $y$-axis

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Surface area of a sphere from $x = r\cos t$, $y = r\sin t$, $0 \le t \le \pi$

31

Speed of a particle on $(x(t), y(t))$

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Polar coordinates $(r, \theta)$

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Polar to rectangular

34

Rectangular to polar

35

Negative $r$ in polar coordinates

36

Convert $(-8, -8)$ to polar

37

Convert $\left(4, \frac{2\pi}{3}\right)$ to rectangular

38

How many polar representations does a point have?

39

$r = 4\sin\theta$ in rectangular form

40

$r = \sec\theta \tan\theta$ in rectangular form

41

Polar line through the pole

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Polar circle through the pole

43

Cardioid

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Limaçon

45

Rose: number of petals of $r = a\cos(b\theta)$ or $a\sin(b\theta)$

46

Archimedean spiral

47

Symmetry about the polar axis

48

Symmetry about the pole

49

Symmetry about the line $\theta = \frac{\pi}{2}$

50

Polar area formula

51

Why is there a $\frac{1}{2}$ in polar area?

52

Area of one petal of $r = 3\sin(2\theta)$

53

Area between two polar curves

54

Area inside $r = 6\sin\theta$ and outside $r = 2 + 2\sin\theta$

55

Hidden intersection of polar curves

56

Polar arc length formula

57

Arc length of the cardioid $r = 2 + 2\cos\theta$

58

Arc length of $r = 3$ (full circle)

59

Slope of a polar curve $r = f(\theta)$

60

Identity for $2 + 2\cos\theta$