254 



PRACTICAL MATHEMATICS 



Simpson's rule is derived from such a curve, the rule can be 

 applied by dividing the area into two strips of equal breadth. 



The breadth of each strip is -. 



41 



Let the ordinates be 



Then 

 where 



v y z , and y 3 . 



The volume = -(y^ + 4y 2 + y s ) 



y^ = area of the base 



y 3 = area of the top 



y 2 = area of the mid-section. 



137. The Centroid. In dealing with a closed, irregular figure, 

 the figure can be enclosed in a rectangle and two adjacent sides 

 of this rectangle can be taken as the axes of reference. The 

 position of the centroid of the figure can then be determined with 

 respect to these axes. 



B 



A X 



FIG. 80. 



Let an irregular figure (Fig. 80) be enclosed in a rectangle OABC, 

 and let the base OA be divided into n equal parts, and the figure 

 divided into n strips of equal breadth by lines drawn through the 

 points of division, perpendicular to OA. Let b be the breadth 

 of each strip, and y lf y 2 , y s . . . y n the mid-ordinates of the strips. 



Taking the first strip and treating it as a rectangle, 



Area of the strip = by 1 



Distance of centroid of the strip from OY = - 



a 



Moment of the strip about OY 



