THE CYCLOID 277 



(a) To find the area. 

 Area - l/y dx 



== \a(l - cos 0) x a(l - cos 6) dQ 

 Jo 



= a- l - 2 cos + cos 2 20) dQ 

 Jo 



= a 2 P'jl - 2 cos + i (1 + cos 20) 



pqfi -i -i2ir 



= a 2 - 2 sin + 7 sin 20 



L2 4 J 



= 3raz 2 



(6) To find the volume of the surface of revolution generated 

 as the area rotates about the base. 



= TT \a*(I - cos 0) 2 x a(l - cos 0) dQ 

 Jo 



1(1 - 3 cos + 3 cos 2 - cos 3 0) dQ 

 Jo 



cos0)|d0 



= Tea 



s - ^r cos + ? cos 20 - 1 cos 30)d0 

 o 12 4 2 4 ) 



[Eft 1 K Q 1 -] 



^ _ f S in + ^ sin 20 - -i sin 30 

 4 4 12 J 



(c) To find the height of the centroid of the cycloidal area 

 above the base. 



By symmetry the centroid is evidently situated in the vertical 

 centre line. Let y be the height. 



Then V ox = 



2;cA 



57T 2 fl 3 



5a 

 "6" 



