

Ex. 2. The sector of a ci 



Let BOE be the sector, sub- 

 ten, liiii: an angle 0, OB a. 



this example we may 

 with equal facility integrate 

 first with respect to and then 

 with respect to r, or first with 

 respect to r and then with re- 

 spect to 6. 



- Kfff cos e dr dd sin ftSd 

 ffitrdrdd 0t 



<W _ (1-cos 



StifrdrdO 



2a(l-cos/3) 



_ 

 ft rdr 



It will be instructive for the student also to notice the 

 solution of this example when rectangular formula 1 arc use.l. 

 The equation to the straight line OE is y = o;tan$; and the 

 equation to the circle EB is x*4-y 2 = a 1 . 



If we integrate with respect to a: first we must integrate 

 from x = y cot $ to x = */(a* ?/) ; since when we i n t 

 with respect to a; we have to collect all the elements in a .-trip 

 which is parallel to the axis of x, and is bounded by 01-' at 

 one end and by EB at the other. These strips extend from 

 the axis of x up to E t and the ordinate of E is a sin /3. 1 1 



with respect to y from y = to # = 

 Therefore 



n*M 

 xdy dx 

 +w 



I 



,J 



"11 



ydydx 





where 



co 



= V (a 1 - 



/*' 



a sn 



The integrations may be easily effected. 



If we wish to integrate with respect to y first, we shall 

 have to divide the figure int its by a straight line 



Irawn from E perpendicular to 01:. l'..r the part to the 



