SEMANTICS OF THE DIFFERENTIAL CALCULUS 593 



The consideration of what is called the definite integral is still more instructive. Let us take the curve in Fig. 12 represented by an equation and a pair of ordinates which intersect the X axis at the points Let us divide the interval x&x_{n} into n 



equal parts and erect ordinates at each point of division. Let us construct a set of pairs of rectangles with these ordinates as we constructed the single pair of rectangles in Fig. 11. By inspection of the figure we see that the area under the curve is slightly greater than 




the sum of the areas of the included rect 
Fig. 12 



angles and slightly less than the sum of
the areas of the including rectangles. When » is allowed to increase without limit the sum of the areas of either set of these rectangles approaches the area bounded by the curve, the X axis, and the end ordinates. In symbols, the area of the first rectangle beneath the curve is wheredenotes 




