![Page 1: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/1.jpg)
Section 6.3: Polar Forms & Area
![Page 2: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/2.jpg)
Rectangular coordinates were horizontal/vertical
directions for reaching the point
Eg: 3, in polar4
Polar coordinates are "as the crow flies"directions
(x,y)
4
3cos ,
3
xso
2 2
1
x = r cos
y = r sin tan
x y r
y
x
Polar Coordinates
![Page 3: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/3.jpg)
Notation
43, 3 3cis 3e4 4 4
i
, r rcis reir
Warnings 5 93, and 3, and 3,
4 4 4
Same point
0, is always the origin
![Page 4: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/4.jpg)
Graph r = 3 Graph =4
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Graph r =
![Page 6: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/6.jpg)
Graph r = 4 sin
r θ
sin
![Page 7: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/7.jpg)
Graph r = 4 sin
r θ
0
sin
![Page 8: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/8.jpg)
Graph r = 4 sin
r θ
0 0
sin
![Page 9: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/9.jpg)
Graph r = 4 sin
r θ
0 0
π/2
sin
![Page 10: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/10.jpg)
Graph r = 4 sin
r θ
0 0
4 π/2
sin
![Page 11: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/11.jpg)
Graph r = 4 sin
r θ
0 0
4 π/2
π
sin
![Page 12: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/12.jpg)
Graph r = 4 sin
r θ
0 0
4 π/2
0 π
sin
![Page 13: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/13.jpg)
Graph r = 4 sin
r θ
0 0
4 π/2
0 π
3π/2
sin
![Page 14: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/14.jpg)
Graph r = 4 sin
r θ
0 0
4 π/2
0 π
-4 3π/2
sin
![Page 15: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/15.jpg)
Graph r = 4 sin
r θ
0 0
4 π/2
0 π
-4 3π/2
2π
sin
![Page 16: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/16.jpg)
Graph r = 4 sin
r θ
0 0
4 π/2
0 π
-4 3π/2
0 2π
sin
![Page 17: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/17.jpg)
r = 2(1 - cos )Cardiod
r θ
![Page 18: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/18.jpg)
r = 2(1 - cos )Cardiod
r θ
0
![Page 19: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/19.jpg)
r = 2(1 - cos )Cardiod
r θ
0 0
![Page 20: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/20.jpg)
r = 2(1 - cos )Cardiod
r θ
0 0
π/2
![Page 21: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/21.jpg)
r = 2(1 - cos )Cardiod
r θ
0 0
2 π/2
![Page 22: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/22.jpg)
r = 2(1 - cos )Cardiod
r θ
0 0
2 π/2
π
![Page 23: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/23.jpg)
r = 2(1 - cos )Cardiod
r θ
0 0
2 π/2
4 π
![Page 24: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/24.jpg)
r = 2(1 - cos )Cardiod
r θ
0 0
2 π/2
4 π
3π/2
![Page 25: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/25.jpg)
r = 2(1 - cos )Cardiod
r θ
0 0
2 π/2
4 π
2 3π/2
![Page 26: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/26.jpg)
r = 2(1 - cos )Cardiod
r θ
0 0
2 π/2
4 π
2 3π/2
2π
![Page 27: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/27.jpg)
r = 2(1 - cos )Cardiod
r θ
0 0
2 π/2
4 π
2 3π/2
0 2π
4
2
2
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r θ
r = 1 2 cosLimacon
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r θ
-1 0
r = 1 2 cosLimacon
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r θ
-1 0
1 π/2
r = 1 2 cosLimacon
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r θ
-1 0
1 π/2
3 π
r = 1 2 cosLimacon
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r θ
-1 0
1 π/2
3 π
1 3π/2
r = 1 2 cosLimacon
3 -1
1
1
![Page 33: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/33.jpg)
r θ
-1 0
1 π/2
3 π
1 3π/2
r = 1 2 cosLimacon
3 -1
1
1
r θ-1 0
1 π/2
![Page 34: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/34.jpg)
r θ
-1 0
1 π/2
3 π
1 3π/2
r = 1 2 cosLimacon
3 -1
1
1
r θ-1 0
1 - √ 2 π/4
1 π/2
![Page 35: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/35.jpg)
r θ
-1 0
1 π/2
3 π
1 3π/2
r = 1 2 cosLimacon
3 -1
1
1
r θ-1 0
1 - √ 2 π/4
0 π/3
1 π/2
![Page 36: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/36.jpg)
r = 4 cos(2 )Rose
r θ
0
π/4
π/2
3π/4
π
5pi/4
3π/2
7π/4
![Page 37: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/37.jpg)
r = 4 cos(2 )Rose
r θ
4 0
π/4
π/2
3π/4
π
5pi/4
3π/2
7π/4
![Page 38: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/38.jpg)
r = 4 cos(2 )Rose
r θ
4 0
0 π/4
π/2
3π/4
π
5pi/4
3π/2
7π/4
![Page 39: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/39.jpg)
r = 4 cos(2 )Rose
r θ
4 0
0 π/4
-4 π/2
3π/4
π
5pi/4
3π/2
7π/4
![Page 40: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/40.jpg)
r = 4 cos(2 )Rose
r θ
4 0
0 π/4
-4 π/2
0 3π/4
π
5pi/4
3π/2
7π/4
![Page 41: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/41.jpg)
r = 4 cos(2 )Rose
r θ
4 0
0 π/4
-4 π/2
0 3π/4
4 π
5pi/4
3π/2
7π/4
![Page 42: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/42.jpg)
r = 4 cos(2 )Rose
r θ
4 0
0 π/4
-4 π/2
0 3π/4
4 π
0 5pi/4
3π/2
7π/4
![Page 43: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/43.jpg)
r = 4 cos(2 )Rose
r θ
4 0
0 π/4
-4 π/2
0 3π/4
4 π
0 5pi/4
-4 3π/2
7π/4
![Page 44: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/44.jpg)
r = 4 cos(2 )Rose
r θ
4 0
0 π/4
-4 π/2
0 3π/4
4 π
0 5pi/4
-4 3π/2
0 7π/4
4
![Page 45: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/45.jpg)
2 r 4cos(2 )Lemniscate 4cos r (2 )Or
r θ
0
π/4
π/2
3π/4
π
5pi/4
3π/2
7π/4
![Page 46: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/46.jpg)
2 r 4cos(2 )Lemniscate 4cos r (2 )Or
r θ
2 0
π/4
π/2
3π/4
π
5pi/4
3π/2
7π/4
![Page 47: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/47.jpg)
2 r 4cos(2 )Lemniscate 4cos r (2 )Or
r θ
2 0
0 π/4
π/2
3π/4
π
5pi/4
3π/2
7π/4
![Page 48: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/48.jpg)
2 r 4cos(2 )Lemniscate 4cos r (2 )Or
r θ
2 0
0 π/4
Undefined π/2
3π/4
π
5pi/4
3π/2
7π/4
![Page 49: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/49.jpg)
2 r 4cos(2 )Lemniscate 4cos r (2 )Or
r θ
2 0
0 π/4
Undefined π/2
0 3π/4
π
5pi/4
3π/2
7π/4
![Page 50: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/50.jpg)
2 r 4cos(2 )Lemniscate 4cos r (2 )Or
r θ
2 0
0 π/4
Undefined π/2
0 3π/4
2 π
5pi/4
3π/2
7π/4
![Page 51: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/51.jpg)
2 r 4cos(2 )Lemniscate 4cos r (2 )Or
r θ
2 0
0 π/4
Undefined π/2
0 3π/4
2 π
0 5pi/4
3π/2
7π/4
![Page 52: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/52.jpg)
2 r 4cos(2 )Lemniscate 4cos r (2 )Or
r θ
2 0
0 π/4
Undefined π/2
0 3π/4
2 π
0 5pi/4
Undefined 3π/2
7π/4
![Page 53: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/53.jpg)
2 r 4cos(2 )Lemniscate 4cos r (2 )Or
r θ
2 0
0 π/4
Undefined π/2
0 3π/4
2 π
0 5pi/4
Undefined 3π/2
0 7π/4
2
![Page 54: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/54.jpg)
1. Something is changing, so we can’t use the old algebra formulas.
2. Break the problem into pieces.
3. Pretend everything is constant on each piece and use the old formulas.
4. Add up the pieces. (This is called a Riemann Sum)
5. If we use more and more pieces, the limit is the right answer! (This limit is a definite integral.)
Big Idea
![Page 55: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/55.jpg)
Insted of breaking the area into little rectangles,
we use little sectors...
Polar Area
That's not surprising if we think of graphing by running
The area of seFact: ctor 2 1is
2r
r
2 1That's easy if you remember the area of circle is 2
2r
![Page 56: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/56.jpg)
Example: Find the area of o rn e = leaf 4 co of s(2 )
Always grImportant: aph first
2
*1Riemann sum
24cos 2 kk
4
2
4
1Integral
24cos 2 d
4 4
![Page 57: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/57.jpg)
4
4
2cos 28 d
4
2
4
1Integral 4cos 2
2d
4
4
11 c8 os 4
2d
2
2
1cos
Two impor
1 cos 22
tant identi
1sin
tie s
1 c s2
:
o 2
x x
x x
4
4
4 1 cos (4 ) d
4
4
14 sin( )4
4
2
![Page 58: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/58.jpg)
Example: Find the area enclosed by and rr = 2s = in 2cos
Find Intersections
= 2 2 s
in cos
But that misses the interaction at origin
because we can only
divide by sin if it's not zero!
!!!Always Graph
tan = 1
5=
4 4
2 - 2
or
r or
![Page 59: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/59.jpg)
4 2
2 2
04
1 12cos2sin
2 2d d
Example: Find the area enclosed by and rr = 2s = in 2cos
![Page 60: Section 6.3: Polar Forms & Area. (x,y) 3 Polar Coordinates](https://reader036.vdocument.in/reader036/viewer/2022062308/56649e3b5503460f94b2daa6/html5/thumbnails/60.jpg)
r = 2 cosExample: Find the area outside the lemniscate
and inside the circl r = 3 c
2
e os
2
: This is not the same as the area
between curves in rectangular coordinates!
You can't do 3cos co1
22 s 2
Warn
d
ing
2 4 22
2 4
1 12 cos 23cos
2 2d d
2 3