VOLUMES OF SURFACES OF REVOLUTION 199 



118. Surfaces of Revolution. When an area rotates about an 

 axis in its plane, it describes a surface of revolution, and the 

 property of such a surface is that any section taken perpendicular 

 to the axis of revolution is circular. 



I 



If the area under a curve (Fig. 47), bounded on the left and 

 right by the ordinates at x = a and x = b respectively, be made 

 to rotate about the axis of x, it describes a surface of revolution, 

 and any section of this surface taken perpendicular to the axis 

 of x will be circular. 



Hence if this area is divided into a very large number of thin 

 strips, each of breadth 8x, each strip will describe a thin cir- 

 cular disc. 



The volume of an elementary disc = 7/ 2 &r. 



The total volume will be obtained by taking the sum of all 

 these elementary discs between the limits x = a and x =*= b. 



\. ^g=6 



Total volume = TC ^ y 2 dx } 



-r z=a 



and, when &r is made infinitely small, 



for the area A, 



ox 



rt 



TC y 2 



Ja 



dx 



If the area B, that bounded by the abscissae which correspond 

 to the ordinates at x = a and x = b, be made to rotate about the 

 axis of y, another surface of revolution is described, and the 

 section of this surface taken perpendicular tc the axis of y will 

 be circular. 



Hence if this area is divided into a very large number of thin 

 strips, each of breadth Sy, each strip will describe a thin circular 

 disc. 



The volume of an elementary disc = rcr 2 8t/. 



