3.6 Numerical integration

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• Approximate the value of a definite integral by using the midpoint and trapezoidal rules.
• Determine the absolute and relative error in using a numerical integration technique.
• Estimate the absolute and relative error using an error-bound formula.
• Recognize when the midpoint and trapezoidal rules over- or underestimate the true value of an integral.
• Use Simpson’s rule to approximate the value of a definite integral to a given accuracy.

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 $f\left(x\right)$ over an interval $\left[a,b\right]$ may be viewed as an estimate of ${\int }_{a}^{b}f\left(x\right)dx.$ Recall that a Riemann sum of a function $f\left(x\right)$ over an interval $\left[a,b\right]$ is obtained by selecting a partition

$P=\left\{{x}_{0},{x}_{1},{x}_{2}\text{,…},{x}_{n}\right\},\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}a={x}_{0}<{x}_{1}<{x}_{2}<\cdots <{x}_{n}=b$

and a set

$S=\left\{{x}_{1}^{*},{x}_{2}^{*}\text{,…},{x}_{n}^{*}\right\},\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}{x}_{i-1}\le {x}_{i}^{*}\le {x}_{i}\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}i.$

The Riemann sum corresponding to the partition $P$ and the set $S$ is given by $\sum _{i=1}^{n}f\left({x}_{i}^{*}\right)\text{Δ}{x}_{i},$ 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.

The midpoint rule

Assume that $f\left(x\right)$ is continuous on $\left[a,b\right].$ Let n be a positive integer and $\text{Δ}x=\frac{b-a}{n}.$ If $\left[a,b\right]$ is divided into $n$ subintervals, each of length $\text{Δ}x,$ and ${m}_{i}$ is the midpoint of the i th subinterval, set

${M}_{n}=\sum _{i=1}^{n}f\left({m}_{i}\right)\text{Δ}x.$

Then $\underset{n\to \infty }{\text{lim}}{M}_{n}={\int }_{a}^{b}f\left(x\right)dx.$

As we can see in [link] , if $f\left(x\right)\ge 0$ over $\left[a,b\right],$ then $\sum _{i=1}^{n}f\left({m}_{i}\right)\text{Δ}x$ corresponds to the sum of the areas of rectangles approximating the area between the graph of $f\left(x\right)$ 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 $\left[a,b\right].$

Using the midpoint rule with ${M}_{4}$

Use the midpoint rule to estimate ${\int }_{0}^{1}{x}^{2}dx$ using four subintervals. Compare the result with the actual value of this integral.

Each subinterval has length $\text{Δ}x=\frac{1-0}{4}=\frac{1}{4}.$ Therefore, the subintervals consist of

$\left[0,\frac{1}{4}\right],\left[\frac{1}{4},\frac{1}{2}\right],\left[\frac{1}{2},\frac{3}{4}\right],\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left[\frac{3}{4},1\right].$

The midpoints of these subintervals are $\left\{\frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8}\right\}.$ Thus,

${M}_{4}=\frac{1}{4}f\left(\frac{1}{8}\right)+\frac{1}{4}f\left(\frac{3}{8}\right)+\frac{1}{4}f\left(\frac{5}{8}\right)+\frac{1}{4}f\left(\frac{7}{8}\right)=\frac{1}{4}·\frac{1}{64}+\frac{1}{4}·\frac{9}{64}+\frac{1}{4}·\frac{25}{64}+\frac{1}{4}·\frac{21}{64}=\frac{21}{64}.$

Since

${\int }_{0}^{1}{x}^{2}dx=\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}|\frac{1}{3}-\frac{21}{64}|=\frac{1}{192}\approx 0.0052,$

we see that the midpoint rule produces an estimate that is somewhat close to the actual value of the definite integral.

Using the midpoint rule with ${M}_{6}$

Use ${M}_{6}$ to estimate the length of the curve $y=\frac{1}{2}{x}^{2}$ on $\left[1,4\right].$

The length of $y=\frac{1}{2}{x}^{2}$ on $\left[1,4\right]$ is

${\int }_{1}^{4}\sqrt{1+{\left(\frac{dy}{dx}\right)}^{2}}\phantom{\rule{0.1em}{0ex}}dx.$

Since $\frac{dy}{dx}=x,$ this integral becomes ${\int }_{1}^{4}\sqrt{1+{x}^{2}}\phantom{\rule{0.1em}{0ex}}dx.$

If $\left[1,4\right]$ is divided into six subintervals, then each subinterval has length $\text{Δ}x=\frac{4-1}{6}=\frac{1}{2}$ and the midpoints of the subintervals are $\left\{\frac{5}{4},\frac{7}{4},\frac{9}{4},\frac{11}{4},\frac{13}{4},\frac{15}{4}\right\}.$ If we set $f\left(x\right)=\sqrt{1+{x}^{2}},$

$\begin{array}{cc}\hfill {M}_{6}& =\frac{1}{2}f\left(\frac{5}{4}\right)+\frac{1}{2}f\left(\frac{7}{4}\right)+\frac{1}{2}f\left(\frac{9}{4}\right)+\frac{1}{2}f\left(\frac{11}{4}\right)+\frac{1}{2}f\left(\frac{13}{4}\right)+\frac{1}{2}f\left(\frac{15}{4}\right)\hfill \\ & \approx \frac{1}{2}\left(1.6008+2.0156+2.4622+2.9262+3.4004+3.8810\right)=8.1431.\hfill \end{array}$

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