AREAS OF SURFACES OF REVOLUTION 278 



QS, and the other whose slant side is CP and the radius of the 

 base is PR. 



Area of elementary surface = 7t(CQ QS - CP PR) 



y)-ly} 

 8s 





By similar triangles 7 = T^ 



and I 8t/ = y 8s 



Hence 8S = 2iiy 8s 



Area of the whole surface = 2ny 8s 



S = 27c|v ds 



ds 

 Hence S = 2* <ir where = 



f* ds 

 = 2* Jj/ ^ <ir, 



f fc ds . ds l~ (dx\ z 



and also S = 2-n: \ y -r- dy, where -j- = A/ 1 + ( -3- ) 

 h dy ' dy V Vrf^y 



Example. Find the area of the surface described by the arc 

 of the curve i/ 2 = 8.r, between the limits x = 2 and # = 4, rotating 

 about the axis of x. 



The limits for y are / = 4 and / = 4\/2. 



Hence 



2 y dy 



