248 



PRACTICAL MATHEMATICS 



133. The Mid-ordinate Rule. If the mid-ordinate is drawn 

 meeting the curve at C (see Fig. 75) , and the tangent ECF is drawn 

 to the curve at that point, the strip may then be taken approxi- 

 mately as a trapezium, the top side of which is the tangent ECF. 

 In this case the lengths of the parallel sides of the trapezium are 

 not known, but it is evident that the mid-ordinate is half the sum 

 of the parallel sides, and therefore the area of the strip is hy{ 

 where y{ is the mid-ordinate. It should be noticed that this is 

 equivalent to taking the strip as being approximately a rectangle 

 the height of which is the mid-ordinate. 



v; 



F 



I 



FIG. 75. 



If the other strips are treated in the same way, the whole area will 

 be approximately equal to the sum of all these equivalent rectangles. 



Area = h(y{ + y' z + y' 3 + . . . y' n ] 



= breadth of strip x sum of the mid-ordinates ... (2) 



For good work it is not safe to use these rules separately, but 

 it is better to take the mean of the results obtained by working 

 with each. The reason for this may be seen from a study of 

 Fig. 75. Using the trapezoidal rule for that strip will give a 

 value for the area in excess of the true value by an amount equal 

 to the area ACB. Using the mid-ordinate rule for the same 

 strip will give a value for the area which is less than the true 

 value by an amount equal to the sum of the areas AEC and BFC. 

 The errors thus involved are opposite in nature, and by taking 

 the mean of the two results there is the tendency for these errors 

 to neutralise each other. 



134. Simpson's Rule for an Odd Number of Ordinates. 



Let the base line be divided into an even number of parts and 

 the ordinates drawn to the curve from the points of division 

 (Fig. 76). The figure is thus divided into an even number of 

 strips of equal breadth. 



