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AREA AND ARC LENGTH INPOLAR COORDINATES
Section 10-5
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Area in Polar Coordinates
The area of the region bounded by the curve between the radial lines
And is given by: )(fr
dfA 2
21
drA 2
21
Handout 10-5 with proofs
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1) Find the area of the region in the plane enclosed by cos12 r
Graph (polar) to find the two radial lines which form the region 20
dfA 2
21
dA 2
0
2cos1221
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1) cont’d
dA 2
0
2cos1221
dA 2
0
2coscos21421
dA
2
0 22cos1cos212
dA 2
0
2cos1cos42
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1) cont’d
dA 2
0
2cos1cos42
dA 2
0
2coscos43
2
022sinsin43
A
6A
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2) Find the area inside the smaller loop of the limacon cos21r
32
34
drA 2
21
dA 3
4
32
2cos2121
dA 3
4
32
2cos4cos4121
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2) cont’d
dA 3
4
32
2cos4cos4121
dA
3
4
32 2
2cos14cos4121
dA 3
4
32
2cos22cos4121
2sinsin4321 3
4
32
A
dA 3
4
32
2cos2cos4321
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2) cont’d 2sinsin4321 3
4
32
A
322sin3
2sin43233
42sin34sin43
4321 A
23
2343
2323
2343
4321
A
334221
A
33221
A
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3) Sketch and set up an integral expression of the area of one petal of )3sin(2 r
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The length of the curve asIs given by:
Arc Length of Polar Curves fr
dddrrL
22
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4) Find the length of the arc from for the cardioid cos22 r
20
dL 2
0
22 sin2cos22
dL 2
0
22 sin4cos4cos84
dL 2
0
22 cossin4cos84
dL 2
0
4cos84
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4) cont’d
dL 2
0
4cos84
dL 2
0
cos88
dL 2
0
cos122
dL 2
0
2
2sin222
dL 2
0 2sin222
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4) cont’d
dL 2
0 2sin222
dL 2
0 2sin4
2
02cos
124
L
16118 L
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5) Find the length of the arc from for using calculator to integrate 2r
50
dL 5
0
222 2
333.63
19L
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