Integration

As mentioned previously, numerical computation of an integral is needed when it is not possible to integrate the expression analytically. In other cases, discrete values of the function may be available at various points. Numerical integration of such functions involves summing up the weighted values of the function evaluated or observed at specified points. The fundamental approach is to construct a trapezoid between any two points, with the two parallel sides being the function values and the interval between the independent variable values constituting the height [4, 10]. If the function is evaluated at two points, a and b, then the following applies:

Decreasing the interval increases the accuracy of the estimate. Several other refinements are also possible but are not discussed here.

Section 4.3 describes various software programs that are available for the computations and solutions of the different types of problems just discussed. These software programs feature built-in tools developed on the basis of these algorithms, obviating any need for an engineer to write a detailed program customized for the problem at hand. The engineer has to know merely how to give the command in the language that is understood by the program. The previous discussion should, however, provide the theoretical basis for the solution as well as illustrate the limitations of the solution technique and possible causes of failure. A course in numerical techniques is often a required core course in graduate chemical engineering programs and sometimes an advanced undergraduate elective course.