THE CENTROID 



215 



119. Considering the area PLKQ (Fig. 55) bounded by the 

 arc PQ of the curve, the axis of y, and the abscissae which corre- 

 spond to the ordi nates at x = a and x = b ; let this area be divided 

 into a large number of thin strips each of breadth By. 



Then area of one strip = x By 

 Moment of the strip = xy By 



since y is the perpendicular distance of the strip from the axis OX. 



Then for the whole area, 



- ji/~t 

 Total moment = > xy By 



K X 



P 



"1 



I 



h 



'1 r - 



FIG, 55, 

 and in the limit when By is made infinitely small, 



Total moment = I xy dy 

 Jh 



Hence Bt/ = I xy dy, where B = I x dy 



Jh Jh 



If the whole area is divided into a very large number of thin 

 strips each of breadth Bx, 



Then area of one strip = (k y) Bx 

 Moment of the strip = (k y}x Bx 



since x is the perpendicular distance of the strip from the 

 axis OY. 



Then for the area PQS, 



^-it-6 



Total moment = > (k y)x dx 



and in the limit when Bx is made infinitely small, 

 Total moment = I (k y)x dx 



Ja 



