![Page 1: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/1.jpg)
AREA AND ARC LENGTH INPOLAR COORDINATES
Section 10-5
![Page 2: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/2.jpg)
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
![Page 3: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/3.jpg)
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
![Page 4: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/4.jpg)
1) cont’d
dA 2
0
2cos122
1
dA 2
0
2coscos2142
1
dA
2
0 2
2cos1cos212
dA 2
0
2cos1cos42
![Page 5: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/5.jpg)
1) cont’d
dA 2
0
2cos1cos42
dA 2
0
2coscos43
2
02
2sinsin43
A
6A
![Page 6: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/6.jpg)
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
![Page 7: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/7.jpg)
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
![Page 8: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/8.jpg)
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
![Page 9: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/9.jpg)
3) Sketch and set up an integral expression of the area of one petal of )3sin(2 r
![Page 10: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/10.jpg)
The length of the curve asIs given by:
Arc Length of Polar Curves fr
dd
drrL
22
![Page 11: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/11.jpg)
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
![Page 12: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/12.jpg)
4) cont’d
dL 2
0
4cos84
dL 2
0
cos88
dL 2
0
cos122
dL 2
0
2
2sin222
dL 2
0 2sin222
![Page 13: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/13.jpg)
4) cont’d
dL 2
0 2sin222
dL 2
0 2sin4
2
02cos
1
24
L
16118 L
![Page 14: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/14.jpg)
5) Find the length of the arc from for using calculator to integrate
2r50
dL 5
0
222 2
333.63
19L
![Page 15: A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5](https://reader037.vdocuments.mx/reader037/viewer/2022103101/56649e3b5503460f94b2da52/html5/thumbnails/15.jpg)
HOME WORKPage 747 # 1-4, 7, 9, 11, 14, 25, 26, 27, 28, 29, 58, 60, 61, 63, 65