278 PRACTICAL MATHEMATICS 



(d) To find the length of the cycloidal curve, 

 dy _ sin 6 

 dx I cos 6 



sin26 



' (1-COS0) 2 

 2(1 - cos 0) 

 (1 - cos 0) 2 



2 

 1 cos 



A/2 



Now ~ , v - . i j- 



\dx 



1 - cos 8) 



A/2af (1 - cos 6)* 

 Jo 



2a 



>o 



6 



p ft-i2w 



4a cos - 



L 2_| 



= 4a{cos 7i cos 0} 



= 8a 



(e) To find the area of the surface of revolution generated as 

 the curve rotates about the base. 



Now -- ' ^ 



(1 - cos 0)' 



f 2 " A/2 



t a(l - cos 0) x - - - , x a(l - cos 

 Jo 1 - cos 8* 



4 (1 - 

 J 



(1 - cos 8) 



cos 



o 

 = 87ia 2 sin 3 : dQ 



f 2 " 6 

 2 sin 3 J: 



Jo 2 



