284 PRACTICAL MATHEMATICS 



l fi 

 Area = - r 2 dQ 



J 



9fi sin 2 0cos 2 0d0 





r- 



(tan 3 + I) 2 



Putting x = tan 



Then dx = sec 2 dQ 

 an 2 sec 2 <ffl f x* dx 

 (tan 3 + I) 2 



Putting x 3 + 1 = y 

 Then dy --= Sx z dx 



f x z dx I Cdy 



and I T r= = - 1-2 



3(tan 3 0+1) 



9f 1 ~ 



= - - - Q 

 2 L 3tan 3 + 1 J 



Hence area Q 



3(tan 3 + 1) J 



EXAMPLES XVIII 



(1) Find the length of the curve y = -(ef* + e~^] between x = 



i 



and x = -. 



(2) Find the area of the surface of revolution produced by that 

 part of the curve y = -(e? + e~ x ) between x = and x = -, rotating 



A a 



about the axis of x. 



