CHAPTER XVII 



132. The work of this chapter will be devoted to the considera- 

 tion of areas, centroids, and moments of inertia of irregular figures. 



The Trapezoidal Rule. Let the base line be divided into a 

 certain number of equal parts and ordinates erected to the curve 

 from the points of division. The area is thus divided into a 

 number of strips of equal breadth, and for n strips there will be 

 n+ 1 ordinates (Fig. 74). 



FIG. 74. 



Let these ordinates be denoted by y v y 2 , y 3 . . . y n+1 



Let h be the breadth of a strip. 



Considering the first strip, an enlarged view of the upper portion 

 of which is shown in Fig. 75 ; by drawing the chord AB the strip 

 may be approximately taken as a trapezium, the area of which 



is + - 



If the other strips are taken in the same way, the whole area 

 will be approximately equal to the sum of all these trapeziums. 



. h. . h. h, h. 



Area = -( Vl + y z ) + -(y z + y 3 ) + -(y 3 + yj + . . 



k " - '. 2/n)} 



-(y n + y n+1 ) 



i(A+*B} 



(1) 



where A = sum of the first and last ordinates 

 and B = sum of the remaining ordinates. 



247 



