100 DIFFERENTIAL AND INTEGRAL CALCULUS. 



AZ7 sinA0 

 therefore - = 1 -- .r (r + Ar), 



p=, or = 



r being a known function of 0. 



(2) Let F be the volume generated by the revolution 

 of AP (fig. 24) about the axis of a?, then A V will be the 

 volume generated by the revolution of area PMNQ about 

 the axis of a?, which will always, when Aa? is small enough, 

 lie between the two cylinders whose common axis is MN, 

 and radii respectively MP, NQ, or between 

 IT '(y*&x) and TT (y + A?/)'' Ax, 



AF 

 or lies between Try 2 and TT Q/ -f 



Aa? 



(3) If 8 be the area of the surface generated by the 

 revolution of AP (fig. 26) about the axis of a?, then A$ 

 will be the area generated by the revolution of the arc 

 PQ or As, and since each point of this arc is at a distance 

 from the axis of a?, which lies between y and y + A?/, 

 therefore AS must lie between 



A5.2?r?/, and As.27r (y + A?/), 



or - lies between 2iry - - and 2?r (?/+ A?/) ; 

 Aa? Aa? A 



therefore 





(4) Let Z7 be the area generated by the revolution 

 of the area AOP (fig. 26) about OA, let fall PJ/ per- 



