3.3.2. The geometric meaning of a definite integral

Let the function y = f ( x ) ≥ 0 be continuous (and therefore integrable) on the segment [ a, b ] (Fig. 13). The integral sum S _n with f ( x ) ≥ 0 is equal to the area of ​​the figure composed of rectangles with sides f (ξ _i ) · h _i . Consequently, the limit of the sequence S _n as h → 0 is equal to the area S of the curvilinear trapezium, i.e. figures bounded by the line y = f ( x ) , axis OX and straight lines x = a, x = b :

Fig. 13