Chapter 11
Parametric Equations and Polar Coordinates
1.
Parameterizations of Plane Curves
2.
Calculus with Parametric Curves
3.
Polar Coordinates
4.
Graphing in Polar Coordinates
5.
Areas and Lengths in Polar Coordinates
6.
Conic Sections
7.
Eccentricity of Conic Sections
1
Lec.4: Lecture Objectives
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Identify what is meant by parametric equations
Graph a parametric curve
Transform Cartesian to parametric equations and vice versa
Obtain derivatives of parametric curves
Evaluate slope, area, length of parametric curves
Evaluate the area of revolution surfaces for parametric curves
Identify what is meant by polar coordinates
Transform polar to cartesian equations and vice versa
Evaluate the slop of polar curves
Graph polar curves
2
Sec. 11.1: Parameterizations of Plane Curves
Parametric Equations:
Ex. Equation of motion of a particle
𝑥 = 𝑓 𝑡 : 𝑥-position
𝑦 = 𝑔 𝑡 : 𝑦-position
𝑡: parameter (time in this example)
3
Ex. Projectile motion
4
Ex. Identify the path traced by the parametric equations:
𝑥 = 𝑡 2 − 2𝑡,
𝑦 =𝑡+1
,0 ≤ 𝑡 ≤ 4
𝑥 = 𝑦 2 − 4𝑦 + 3 Cartesian eqn.
5
Ex. Identify geometrically the curve by eliminating the
parameter:
𝑥 = 𝑐𝑜𝑠 𝑡 ,
𝑦 = 𝑠𝑖𝑛 𝑡 ,
0 ≤ 𝑡 ≤ 2𝜋
Cartesian eqn.
𝑥2 + 𝑦2 = 1
(Unit Circle)
6
Ex. Identify geometrically the curve by obtaining the
Cartesian equation:
𝑥 = 3𝑐𝑜𝑠 𝜃 ,
𝑦 = 4𝑠𝑖𝑛 𝜃 ,
0 ≤ 𝜃 ≤ 2𝜋
Cartesian eqn.
𝑥2 𝑦2
+
=1
9 16
(Ellipse)
7
Ex. Find the Cartesian equation for the cycloid,
𝑥 = 𝑎 𝑡 − 𝑠𝑖𝑛 𝑡 ,
𝑦 = 𝑎 1 − 𝑐𝑜𝑠 𝑡
,𝑡 ≥ 0
http://en.wikipedia.org/wiki/Cycloid
8
Ex. Find the Cartesian equation for the cycloid,
𝑥 = 𝑎 𝑡 − 𝑠𝑖𝑛 𝑡 ,
𝑦 = 𝑎 1 − 𝑐𝑜𝑠 𝑡
,𝑡 ≥ 0
en.wikipedia.org/wiki/Tautochrone_curve
9
Ex. Find the Cartesian equation for the cycloid,
𝑥 = 𝑎 𝑡 − 𝑠𝑖𝑛 𝑡 ,
𝑦 = 𝑎 1 − 𝑐𝑜𝑠 𝑡
,𝑡 ≥ 0
10
Sec. 11.2: Calculus with Parametric Curves
11
Ex: Find
at 𝑡 =
𝑑𝑦 𝑑 2 𝑦
,
𝑑𝑥 𝑑𝑥 2
𝜋
:
4
and the equation of the tangent to the curve
𝑥 = 𝑠𝑒𝑐 𝑡 , 𝑦 = 𝑡𝑎𝑛 𝑡
𝜋
,−
2
<𝑡<
𝜋
2
𝑑𝑦 𝑑𝑦/𝑑𝑡 𝑠𝑒𝑐 𝑡
=
=
= 𝑐𝑠𝑐 𝑡
𝑑𝑥 𝑑𝑥/𝑑𝑡 𝑡𝑎𝑛 𝑡
𝑑𝑦
ቤ
= 2,
𝑑𝑥 𝑡=𝜋/4
Tangent line: 𝑦 − 1 = 2 𝑥 − 2
𝑑 2 𝑦 𝑑𝑦′/𝑑𝑡 −𝑐𝑠𝑐 𝑡 𝑐𝑜𝑡 𝑡
=
=
2
𝑑𝑥
𝑑𝑥/𝑑𝑡
𝑠𝑒𝑐 𝑡 𝑡𝑎𝑛 𝑡
𝑑2𝑦
= −1
อ
2
𝑑𝑥
𝑡=𝜋/4
12
Area under a Parametric Curve
Ex: Find the area under one arch of the Cycloid:
𝑥 = 𝑎 𝑡 − 𝑠𝑖𝑛 𝑡 ,
𝑦 = 𝑎 1 − 𝑐𝑜𝑠 𝑡
, 0 ≤ 𝑡 ≤ 2𝜋
𝑥2
𝐴 = න 𝑦 𝑑𝑥
𝑥1
2𝜋
= 𝑎2 න 1 − 𝑐𝑜𝑠 𝑡
2 𝑑𝑡
0
= 3𝜋𝑎2
13
Length of Parametric Curves
𝑑𝑙 =
𝑑𝑙 =
𝑑𝑥
𝑑𝑥
𝑑𝑡
2
2
+ 𝑑𝑦
𝑑𝑦
+
𝑑𝑡
2
2
𝑑𝑡
14
Ex: Find the length of the Astroid:
𝑥 = cos3 𝑡 ,
𝑦 = sin3 𝑡
, 0 ≤ 𝑡 ≤ 2𝜋
𝜋/2
𝐿 = 12 න 𝑠𝑖𝑛 𝑡 𝑐𝑜𝑠 𝑡 𝑑𝑡 = 6
0
en.wikipedia.org/wiki/Astroid
15
Area of Surface of Revolution of Parametric Curves
16
Ex: The standard parametrization of the circle of radius 1
centered at the point 0,1 in the 𝑥𝑦-plane is:
𝑥 = 𝑐𝑜𝑠 𝑡 ,
𝑦 = 1 + 𝑠𝑖𝑛 𝑡
, 0 ≤ 𝑡 ≤ 2𝜋
Find the surface area of the solid generated by revolving the
circle about the 𝑥-axis
𝑑𝑙 =
(− sin 𝑡) 2 +(cos 𝑡) 2 𝑑𝑡 = 𝑑𝑡
2𝜋
𝑆 = 2𝜋 න 1 + 𝑠𝑖𝑛 𝑡 𝑑𝑡
=
0
4𝜋 2
17
Sec. 11.3: Polar Coordinates
In many cases,
polar coordinates are simpler ,
easier and more convenient to use than
cartesian (rectangular) coordinates.
18
Polar Coordinates
Terminal Ray
19
Cartesian grid
Polar grid
20
21
The center of the graph is
called the pole.
Points are
represented by a
radius and an angle
radius
(r, )
To locat the point
 
 5, 
 4
First find the angle
Then move along this
direction 5 units
P (r, )
r is directed distance
+ve 
 is directed angle
-ve 
P (-r, )
-∞ < r < ∞
-∞ <  < ∞
(r, ) = (r, ±2nπ)
Locate  first then r.
23
Polar coordinates are not unique
24
Locate the point (2, 7π/6)
25
Relation between Polar Coordinates and Cartesian
Coordinates
26
Ex. Find polar coordinates of the Cartesian point
P: (−2, −2 3)
r2
= −2
tan 𝜃 =
2
+ −2 3
−2 3
−2
2
= 16
= 3
Polar coordinates of P is
OR
4𝜋
(4, )
3
𝜋
(−4, )
3
27
Important Polar Curves
𝑥 2 + 𝑦 2 = 16
𝑦=𝑥
28
Ex. Graph the region (set of points) satisfying:
𝜋
1 ≤ 𝑟 ≤ 2,
0≤𝜃≤
2
29
Ex. Graph the region (set of points) satisfying:
2𝜋
5𝜋
≤𝜃≤
3
6
30
The line 𝑥 = 3
𝑟 𝑐𝑜𝑠 𝜃 = 3
The line 𝑦 = 2
𝑟 𝑠𝑖𝑛 𝜃 = 2
31
Ex. Find a polar equation for the circle:
𝑥2 + 𝑦 − 3 2 = 9
𝑥 2 + 𝑦 2 − 6𝑦 = 0
𝑟 2 − 6𝑟𝑠𝑖𝑛 𝜃 = 0
𝑟 = 6 𝑠𝑖𝑛 𝜃
32
2
𝑥 + 𝑦−𝑎
2
=𝑎
𝑟 = 2𝑎 𝑠𝑖𝑛 𝜃
2
(𝑥 − 𝑎)2 +𝑦 2 = 𝑎2
𝑟 = 2𝑎 𝑐𝑜𝑠 𝜃
33
Sec. 11.4: Graphing in Polar Coordinates
Slope of tangent line to the curve 𝑟 = 𝑓(𝜃)
𝑥 = 𝑟 𝑐𝑜𝑠 𝜃,
𝑦 = 𝑟 𝑠𝑖𝑛 𝜃
𝑑𝑦
𝑑𝑟
sin 𝜃 + 𝑟 𝑐𝑜𝑠 𝜃
𝑑𝑦
𝑑𝜃
𝑑𝜃
=
=
𝑑𝑥
𝑑𝑟
𝑑𝑥
𝑐𝑜𝑠 𝜃 − 𝑟 𝑠𝑖𝑛 𝜃
𝑑𝜃
𝑑𝜃
𝑓′ sin 𝜃 + 𝑓 𝑐𝑜𝑠 𝜃
=
𝑓′ 𝑐𝑜𝑠 𝜃 − 𝑓 𝑠𝑖𝑛 𝜃
34
Ex. Find the slope of the tangent line to the circle
𝑟 = 4𝑐𝑜𝑠 𝜃 at the point where 𝜃 = 𝜋/4
𝑑𝑟
= −4 sin θ
𝑑𝜃
𝑑𝑦
−4 𝑠𝑖𝑛2 𝜃 + 4 cos 2 𝜃
=
𝑑𝑥 −4 sin 𝜃 𝑐𝑜𝑠 𝜃 − 4 𝑐𝑜𝑠 𝜃 sin 𝜃
𝑑𝑦
ቤ
=0
𝑑𝑥 𝜃=𝜋/4
35
Symmetry in Polar Coordinates
Symmetry of the curve 𝑟 = 𝑓(𝜃)
1. about 𝑥 − 𝑎𝑥𝑖𝑠
𝑟, 𝜃 ⇒ 𝑟, −𝜃 or −𝑟, 𝜋 − 𝜃
𝑟 = 2 cos 𝜃
36
Symmetry of the curve 𝑟 = 𝑓(𝜃)
2. about 𝑦 − 𝑎𝑥𝑖𝑠
𝑟, 𝜃 ⇒ −𝑟, −𝜃 𝑜𝑟 𝑟, 𝜋 − 𝜃
𝑟 = 𝑠𝑖 𝑛 𝜃
37
Symmetry of the curve 𝑟 = 𝑓(𝜃)
3. about the origin
𝑟, 𝜃 ⇒
−𝑟, 𝜃 or 𝑟, 𝜃 + 𝜋
𝑟 2 = 𝑠𝑖𝑛(2𝜃)
(Lemniscate)
38
Lemniscate Antenna
39
Check the Symmetry
about x-axis
𝑟, 𝜃 ⇒ 𝑟, −𝜃
𝑟, 𝜃 ⇒ −𝑟, 𝜋 − 𝜃
about y-axis
𝑟, 𝜃 ⇒ −𝑟, −𝜃
𝑟, 𝜃 ⇒ 𝑟, 𝜋 − 𝜃
about origin
𝑟, 𝜃 ⇒ −𝑟, 𝜃
𝑟, 𝜃 ⇒ 𝑟, 𝜃 + 𝜋
Ex. r = sin 2𝜃
40
Ex. Transform the equation of the following curve to polar
coordinates and graph it: 𝑥 2 + 𝑦 2 + 𝑥 = 𝑥 2 + 𝑦 2
𝑟 = 1 − cos 𝜃
symmetric about 𝑥 − 𝑎𝑥𝑖𝑠
Cardioid
41
Ex. Transform the equation of the following curve to polar
coordinates and graph it: 𝑥 2 + 𝑦 2 + 𝑥 2 = 𝑥 2 + 𝑦 2
𝑟 = 1 − cos 𝜃
r
2
1
-3
-2
-1
1
2
3
4
5
6
7
8
9

Cardioid
𝑟 = 𝑎 1 + 𝑐𝑜𝑠 𝜃
𝑟 = 𝑎 1 − 𝑐𝑜𝑠 𝜃
𝑟 = 𝑎 1 + 𝑠𝑖𝑛 𝜃
𝑟 = 𝑎 1 − 𝑠𝑖𝑛 𝜃
A cardioid is a curve traced by a point on the perimeter
of a circle that is rolling around a fixed circle of the
same radius.
en.wikipedia.org/wiki/Cardioid
44
𝑈𝑛𝑖 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑀𝑖𝑐𝑟𝑜𝑝ℎ𝑜𝑛𝑒
45
Pick up Pattern
46
𝑂𝑚𝑛𝑖 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑀𝑖𝑐𝑟𝑜𝑝ℎ𝑜𝑛𝑒
47
Spirals:
Archimedean Spiral Logarithmic Spiral
𝑏𝜃
𝑟
=
𝑎𝑒
𝑜𝑟
𝑟 = 𝑎𝜃
1 𝑟
𝜃 = 𝑙𝑛
𝑏 𝑎
48
Archimedean Spiral
Hamilton Watch
A Sailor’s coiled rope
49
Rose Curves
50
Find the Cartesian Equation:
51
Find the slope of the curve:
52
53
54
55
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