FBI Of CKAVITV. 



of irr.-ivity of tlio volume generated by that half of the 

 curve for which y is positive. 1 1 ere 



Now fx*ydy ^fz'y-J'dz'y thus in tin- present case 

 J a?ydy=j x*y -j^dx. 



Similarly 



Tin, 



MV)^ /o * 



The numerator of y consists of two integrals which 

 the following values, 



I x (2ax x*)dx = , 

 Jo 6 



a I xJ(2ax x*) vers' 1 - <c = a ( --H ). 



Jo a \9 A / 



The value of the denominator of y is a 3 . 



4a 4 4a 4 



T^T" 1 "! 



Therefore y = 



AVe may also find it convenient in some cases t 

 formulae involving double integration. 



Suppose the figure in Art. 118 to revolve round the axil 

 of x; let a;, y be the co-ordinates of x; and x+ Ao-, y -f Ay 

 those of *. The area generates by revolution an el< UK utary 

 ring, the volume of which is TT (y + Ay) 2 Ax - Try'Aa: ; this 

 may be put ultimately equal to ^TryAyAsr. The cent 

 gravity of this ring is on the axis of x, and its alj.-ci- 



