260 



PRACTICAL MATHEMATICS 



Let the length of one of these strips be AB, the perpendicular 

 distance of this strip from the axis OX being y. 



The area of the strip = AB &?/ 



The moment of the strip about OX = ABt/ $y 



Let QT be the projection of AB on LN and the lines PQ and 

 PT drawn cutting AB in A x and B x respectively. 



FIG. 82, 



Then if h is the height of the rectangle, 



QT 



_, 



By similar triangles -T- = 



or 



and 



T 



yQT=h A 

 y AB = h A 

 y AB $y = h Aj 



since AB = QT 



Hence the moment of the strip about OX = ^AjBj y. 

 The moment of the irregular area about OX would be obtained 

 by taking the sum of the moments of all these strips, 



M ox = SAB y 



and 



Now EAjBj $y is the area of the figure obtained by joining 

 all the points derived in the same manner as A x and Bj for 

 different positions of the horizontal line AB between the limits 

 y = and y = h. This figure is spoken of as the " first derived 

 figure." 



Then M ox = h x area of the first derived figure 



