334 



SUKFACE OF A SOLID OF REVOLUTION. 



Let V denote the volume of a portion of this solid contained 

 between two planes perpendicular to the axis Ox, one drawn 

 through a fixed point A and the other through P. Let A V 

 denote the volume of the solid contained between planes 

 through P and Q perpendicular to the axis. The volume 

 of a cylinder having MN for its axis and for its base the 

 circular area formed by the revolution of PM round the axis 

 Ox, is Try*&x. The volume of a cylinder having MN for its 

 axis and for its base the circular area formed by the revolu- 

 tion of QN round Ox, is IT (y + &yf Arc. Hence A V lies 



AF 

 between Tn/'Ao; and 7r(y+ A#) 2 A#. Therefore -^ lies be- 



tween Try 2 and TT ( y + Ay) 2 . Hence, proceeding to the limit, 

 we have 



dV 



315. Differential coefficient of the surface of a solid of re- 

 volution. 



Let P, Q, be two points in a curve which by revolving 

 round the axis Ox generates 

 a solid of revolution. Let A 

 be a fixed point on the curve, 

 and suppose AP = s, PQ= A*. 

 Let 8 denote the area of the 

 surface formed by the revolu- 

 tion of AP, and A$ the area 

 of the surface formed by the 

 revolution of PQ. Draw PR and Q T each equal to As and 

 each parallel to Ox. If PR revolve round Ox it generates 

 a cylinder, the surface, of which is 2-TryAs. If QT revolve 

 round Ox it generates a cylinder, the surface of which is 

 27r (y + A^) As. We assume as an axiom that the surface 

 generated by the arc PQ lies between the former and the 

 latter. Hence A$ lies between 2-TryAs and 27r(^+ A?/) As, 

 and proceeding to the limit, we have 



dS 



therefore 





 dx 







dx 



