3.6 Numerical integration

The antiderivatives of many functions either cannot be expressed or cannot be expressed easily in closed form (that is, in terms of known functions). Consequently, rather than evaluate definite integrals of these functions directly, we resort to various techniques of numerical integration    to approximate their values. In this section we explore several of these techniques. In addition, we examine the process of estimating the error in using these techniques.

The midpoint rule

Earlier in this text we defined the definite integral of a function over an interval as the limit of Riemann sums . In general, any Riemann sum of a function $f(x)$ over an interval $[a,b]$ may be viewed as an estimate of ${\displaystyle {\int }_a^{b}f(x)dx}.$ Recall that a Riemann sum of a function $f(x)$ over an interval $[a,b]$ is obtained by selecting a partition

and a set

The Riemann sum corresponding to the partition $P$ and the set $S$ is given by ${\displaystyle \sum _{i=1}^{n}f(x_i^{*})\text{Δ}{x}_i},$ where $\text{Δ}{x}_i=x_i-x_{i-1},$ the length of the i th subinterval.

The midpoint rule    for estimating a definite integral uses a Riemann sum with subintervals of equal width and the midpoints, $m_i,$ of each subinterval in place of $x_i^{*}.$ Formally, we state a theorem regarding the convergence of the midpoint rule as follows.

As we can see in [link] , if $f(x)\ge 0$ over $[a,b],$ then ${\displaystyle \sum }_{i=1}^{n}f(m_i)\text{Δ}x$ corresponds to the sum of the areas of rectangles approximating the area between the graph of $f(x)$ and the x -axis over $\left[a,b\right].$ The graph shows the rectangles corresponding to $M_4$ for a nonnegative function over a closed interval $[a,b].$