VOLUMES OF SURFACES OF REVOLUTION 208 



fZO 



For the area B, V oy = TC I x 2 dy 



-|JV* 

 --[>T 



v>L y l 



120 

 J& 



= 37-57T = 117-8 



20 



10 



B 



FIG, 49. 



r fZQ -v 



For the area A, V oy = TC { 20(2) 2 - 5(1) 2 - \ x z dy \ 



= TC {80- 5 -37-5} 

 = 37-57T = 117-8 



For the area B, V ox = n J2(20) 2 - 1(5) 2 - f^ 2 dx\ 



= 7r{800-25- 155} 

 = 620TC = 1947 



Example 2. In the curve y = e~*, taking A as the area bounded 

 by the ordinates at x = 1 and x = 2, and B as the area bounded 

 by the abscissae corresponding to the ordinates at x = 1 and # = 2, 

 find the volumes of the surfaces of revolution : 



(1) when the area A rotates about the axes of x and y respec- 



tively ; 



(2) when the area B rotates about the axes of x and y respec- 



tively. 



