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< CHAPTER 21. Numerical Integration | Contents | 21.2 Riemann’s Integral >

Numerical Integration Problem Statement

Given a function \(f(x)\), we want to approximate the integral of \(f(x)\) over the total interval, \([a,b]\). The following figure illustrates this area. To accomplish this goal, we assume that the interval has been discretized into a numeral grid, \(x\), consisting of \(n+1\) points with spacing, \(h = \frac{b - a}{n}\). Here, we denote each point in \(x\) by \(x_i\), where \(x_0 = a\) and \(x_n = b\). Note: There are \(n+1\) grid points because the count starts at \(x_0\). We also assume we have a function, \(f(x)\), that can be computed for any of the grid points, or that we have been given the function implicitly as \(f(x_i)\). The interval \([x_i, x_{i+1}]\) is referred to as a subinterval.

Illustration integral

The following sections give some of the most common methods of approximating \(\int_a^b f(x) dx\). Each method approximates the area under \(f(x)\) for each subinterval by a shape for which it is easy to compute the exact area, and then sums the area contributions of every subinterval.

< CHAPTER 21. Numerical Integration | Contents | 21.2 Riemann’s Integral >