4Numerical Quadrature


Definite integrals must be evaluated routinely and for cases in which an antiderivative cannot be found, or for cases in which the analytic process is simply too difficult, a numerical scheme (numerical integration, or quadrature) may be our only recourse. Consider the integral

which we know to have the value c4-math-5001. Let us consider a plot of the integrand as a function of x, shown in Figure 4.1.

Figure 4.1.  Behavior of x2 exp(−x2). The value of the integral, eq. (4.1), is known to be c4-math-5061.

We will conduct an elementary experiment with the graph of this integrand: First, we will count the smaller rectangular regions under the curve and add their areas together to obtain (70)(0.25)(0.025) = 0.4375. Next, we will cut out the region under the curve and weigh it and compare that weight to the average weight per rectangle. For the paper under the curve, we find W = 0.5754 g, and for each box, 0.008003 g. Therefore, the area under the curve is approximately 71.89 rectangular boxes or (71.89)(0.25)(0.025) = 0.4493. Our first estimate is in error by about 1.3% ...

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