VOLUME OF A SOLID OF REVOLUTION. 333 



r cos 

 = Ar+2r sin 2 . 



m 



Also the chord PQ = J(PL* + 



From this, if we proceed according to the method of the 

 preceding Articles, we shall arrive at 



ds _ I ( 2 (da 



7/J A' / "l* T I 7/ 



312. If A denote the area contained between a curve and 

 the axis, of x, we have shewn in Art. 43 that 



dA 



313. To find the differential coefficient of the area of a 

 curve referred to polar co-ordinates. 



Let A denote the area contained between the radius /SP ? 

 the radius SC drawn to some 

 fixed point on the curve, and 

 the curve CP. Let &.A denote 

 the area PSQ. With centre S 

 and radius SP describe an arc 

 meeting SQ at L, and with 

 centre S and radius SQ describe 

 an arc meeting SP produced at 

 M. Then AJ. lies between PSL and QSM, that is, between 



,1 f r i, j 



therefore -r^ lies between and . 



Hence, proceeding to the limit, we have 

 dA 9* 

 d6~ 2' 



314. Differential coefficient of the volume of a solid of re- 

 volution. 



Suppose the curve APQ in the figure of Art. 307 to 

 revolve round the axis of x, and thus to generate a solid. 



