214 PRACTICAL MATHEMATICS 



and in the limit when &r is made infinitely small, 



f & 

 Total moment = I xy dx 



Ja 



Hence Aid = I xy dx, where A = I y dx 



Ja Ja 



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

 strips each of breadth y 



Then area of one strip = (b x} y 

 Moment of the strip = (b x)y $y 



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

 Then for the area PQR, 



Total moment = > (b x}y ?>y 

 ^- 'v=fc 



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



<4 



Total moment = I (b x}y dy 



Jh 



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

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

 the moment of the whole area PMNQ. 



Area of rectangle = h(b a) 



Moment of rectangle = - h z (b a) 



it 



6 



dx 



c k i r 



Hence Ay = I (b x}y dy + -h z (b a), where A = I y 



h ^ Ja 



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

 as being divided up into thin vertical strips, each of breadth 8#. 



Area of one strip = y $x 

 Moment of strip = -y 2 $x 







since -y is the perpendicular distance of the centroid of this 



* 



strip from the axis of x. 



Then for the whole area PMNQ, 



1 



Total moment = - > y 2 



and in the limit when S# is made infinitely small, 



1 f 6 

 Total moment = -- \ y 2 dx 



2 Jo 



1 f & f 6 



Hence Ay = - \ y* dx, where A = \y dx 



* Ja Ja 



