THE CENTROID 



Tabulating these results : 



213 



118. The position of the centroid of a given area may be found 

 by direct integration. It has already been shown that if x and y 

 are the co-ordinates of the centroid, then 



and 



x = 



H 



or, in other words, 



Aa? = sum of the moments of all the elementary areas taken 

 with respect to the axis of y, 



and A.y = sum of the moments of all the elementary areas taken 

 with respect to the axis of x. 



FIG, 54. 



Considering the area PMNQ (Fig. 54) bounded by the arc PQ 

 of a curve, the axis of x, and the ordinates at x = a and x = b. 

 Let this area be divided into a large number of thin strips each 

 of breadth 8x. 



Then area of one strip = y 8x 



Moment of the strip = xy x, since x is the perpendicular 

 distance of the strip from the axis OY. 



Then for the whole area, 



Total moment = 





I 



xy 



