216 PRACTICAL MATHEMATICS 



This only gives the moment of the area PQS, and to this must 

 be added the moment of the rectangle PLKS, in order to obtain 

 the moment of the whole area PLKQ. 



Area of rectangle = a(k h) 



Moment of rectangle = -o?(k h) 

 2 



j* 1 f fc 



Hence Ba? = I (k y)x dx + ~a 2 (A; h), where B = I x dy 



Ja h 



The expression for x could also be obtained by taking the area 

 as being divided into thin horizontal strips each of breadth 8/. 

 Area of one strip = x y 



Moment of strip = -x 2 % 



Zt 



since -x is the perpendicular distance of the centroid of this strip 



m 



from the axis of y. 



Then for the whole area PLKQ, 



1 V l v ' fc 

 Total moment = - > x 2 $ 



and in the limit when %y is made infinitely small, 



1 f k 

 Total moment = - I x 2 dy 



* Jh 



Hence B# = - I x 2 dy, where B = I x dy 



2 h h 



As an illustration of the application of this method of finding 

 the position of the centroid of an area, let us take the curve of 

 the previous example, y = 2-273# r137 , and work, as before, with 

 x = 2 and x = 4 as the limits for x, and y = 5 and y = 11 for the 

 corresponding limits of y. 



It has already been shown that area A = 15-92 and area 

 B = 18-08. 



(1) For the area A, and taking vertical strips each of breadth &c. 



Moment of strip = xy %x 



f 4 



= I xy dx 



J2 



dx 



Total moment 



= a\ x n - 

 Ji 



o.o7? 



= _ _ /43. 137 _ 03-137X 



3-137 l 

 = 49-72 



