DIFFERENTIAL AND INTEGRAL CALCULUS. 101 



pendicular on OA, and produce MP to meet SQ in R, 

 then SP = r, AOP=6, SQ = r+&r, ASQ=0 + &0, and 

 the volume generated by the revolution of POQ is SZ7, 

 but the volume generated by the area PQR vanishes in 

 the limit compared with that generated by POR, hence 

 A ~U ultimately the volume generated by POQ volume 

 generated by MOR volume generated by MOP 



shi'tf.r cos#-h JTT {r cos 0. tan (0 + A0)}V cos# 

 cos0(- sin 2 6 + cos 2 6 tan 2 0') 



cos' 6 



Q sin (2(9 

 therefore .. 



dU sin20 



therefore U = | TT f * r 3 sin 6W, 



-'o 



r being a known function of 6. 



As an example of the use of all these formulae, take the 

 curve 9y 2 =(x + a) (x 2a) 2 , which is of the form given, 

 (fig. 27) 



dy _ x (x 2a) _ a? (a? 2a) 



dx~ bay ~ 2 V{ ( + a) (a? - 

 < ic a; 





2 V{(^ + )} 2 V 

 if we take the lower branch BAP, or 



x z / x V 



therefore sin' 2 \//= 7 r- 5 = - 



4a (a; -f a) + ar \5C + 2aJ 7 



