AREA AND ARC LENGTH INPOLAR COORDINATES
Section 10-5
Area in Polar Coordinates
The area of the region bounded by the curve between the radial lines
And is given by: )(fr
dfA 2
2
1
drA 2
2
1
Handout 10-5 with proofs
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
2
1
dA 2
0
2cos122
1
1) cont’d
dA 2
0
2cos122
1
dA 2
0
2coscos2142
1
dA
2
0 2
2cos1cos212
dA 2
0
2cos1cos42
1) cont’d
dA 2
0
2cos1cos42
dA 2
0
2coscos43
2
02
2sinsin43
A
6A
2) Find the area inside the smaller loop of the limacon cos21r
32
34
drA 2
2
1
dA 3
4
32
2cos212
1
dA 3
4
32
2cos4cos412
1
2) cont’d
dA 3
4
32
2cos4cos412
1
dA
3
4
32 2
2cos14cos41
2
1
dA 3
4
32
2cos22cos412
1
2sinsin432
13
4
32
A
dA 3
4
32
2cos2cos432
1
2) cont’d 2sinsin432
13
4
32
A
322sin3
2sin43233
42sin34sin43
432
1 A
2
3
2
343
232
3
2
343
432
1
A
33422
1 A
3322
1 A
3) Sketch and set up an integral expression of the area of one petal of )3sin(2 r
The length of the curve asIs given by:
Arc Length of Polar Curves fr
dd
drrL
22
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
4) cont’d
dL 2
0
4cos84
dL 2
0
cos88
dL 2
0
cos122
dL 2
0
2
2sin222
dL 2
0 2sin222
4) cont’d
dL 2
0 2sin222
dL 2
0 2sin4
2
02cos
1
24
L
16118 L
5) Find the length of the arc from for using calculator to integrate
2r50
dL 5
0
222 2
333.63
19L
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