200 PRACTICAL MATHEMATICS 



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

 these elementary discs between the limits y = h and y = k. 



Total volume = TT 



and, when y is made infinitely small, 



ffc 

 for the area B, V OY = TT! x 2 dy 



h 



When the area A rotates about the axis of y, the surfaces of 

 revolution described by the rectangle DQHO, the rectangle 

 EPGO and the area EPQD must be considered. 



The rectangle DQHO describes a cylinder of volume 7ib 2 k, the 

 rectangle EPGO describes a cylinder of volume 7ta 2 h, and the 



rk 

 area EPQD describes a surface of revolution of volume TT| x 2 dy. 



1* 



Hence for the area A, V OY = Ttb 2 k 7ta 2 h TT I x 2 dy 



h 

 = ii(b 2 k - a 2 h - f x 2 dy\ 



When the area B rotates about the axis of x, the surfaces of 

 revolution described by the rectangle DQHO, the rectangle 

 EPGO and the area PQHG must be considered. 



The rectangle DQHO describes a cylinder of volume 7tk 2 b, the 

 rectangle EPGO describes a cylinder of volume Tch 2 a, and the 



fb 



area PQHG describes a surface of revolution of volume ?r I y 2 dx. 



Ja 



n 

 Hence for the area B, V ox = nk 2 b - Tih 2 a TT I y 2 dx 



Ja 



= nfk 2 b -h 2 a- f 6 y 2 dx\ 



As an example, let the law of the curve be y = cx n where c and n 

 are constants. 



n 



Then for the area A, V ox = iz 1 y 2 dx 



Ja 



x 



= TO 



2n dx 



2n + l 

 Now h = ca n and k = cb n 



