POLAR CO-ORDINATES 281 



The area of the sector OAB (working with Oj and 2 as the 

 limits for 6) will be the sum of the areas of all these elementary 

 sectors. 



Area of sector AOB - > - r 2 86 



SD 



when 80 is made infinitely small. 



Tojind the length of the curve. 

 For a very small arc PQ 



PQ 2 = QR 2 + PR 2 

 That is 8s 2 = 8r 2 + r 2 80 2 



Then 



or 



When 80 becomes infinitely small, 

 ds 



or 



Hence to ^et s, the length of a certain part of the curve, the 

 first expression must be integrated with respect to and the 

 second with respect to r ; in each case the integration being taken 

 between given limits, 





- 



or * = 



149. Example I. To find the area and the length of the 

 cardioid, the equation of which is r = a(l cos 0). 



