Graphical, Numerical, and Mathematical Methods

Often the mathematical function /(x) to be integrated is too complex and we are not able to integrate it analytically. Or in some cases the function is one that has been obtained from experimental data, and no mathematical equation is available to represent the data so that they can be integrated analytically. In these cases, we can use graphical integration. 

Integration between the limits x — a to x = b can be represented graphically as shown in Fig. 1.8-1. Here a function y = f(x) has been plotted versus x. The area under the curve y = f x) between the limits x = n to x = b is equal to the integral. This area is then equal to the sum of the areas of the rectangles as follows. 

Often it is desired or necessary to perform a numerical integration by computing the value of a definite integral from a set of numerical values of the integrand /(x). This, of course, can be done graphically, but if data are available in large quantities, numerical methods suitable for the digital computer are desired. 

The value of h, a constant, is the spacing in x used. Then, approximating /(x) by a parabola on each subinterval, Simpson s rule is 

Temperature of a ChemicaJ Process, The temperature of a chemical reaction was 

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