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

HOME WORKPage 747 # 1-4, 7, 9, 11, 14, 25, 26, 27, 28, 29, 58, 60, 61, 63, 65