6.1: Elementary Formulas

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May 31, 2022
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- 6: Integration
- 6.2: Composite Rules

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We first consider integration from 0 to $h$ , with $h$ small, to serve as the building blocks for integration over larger domains. We here define $I_{h}$ as the following integral:

$I_{h}=\int_{0}^{h} f(x) d x . \nonumber$

To perform this integral, we consider a Taylor series expansion of $f(x)$ about the value $x=h / 2$ :

$\begin{aligned} f(x)=f(h / 2)+(x-h / 2) f^{\prime} &(h / 2)+\frac{(x-h / 2)^{2}}{2} f^{\prime \prime}(h / 2) \\ &+\frac{(x-h / 2)^{3}}{6} f^{\prime \prime \prime}(h / 2)+\frac{(x-h / 2)^{4}}{24} f^{\prime \prime \prime \prime}(h / 2)+\ldots \end{aligned} \nonumber$

6.1.1. Midpoint rule

The midpoint rule makes use of only the first term in the Taylor series expansion. Here, we will determine the error in this approximation. Integrating,

$\begin{aligned} I_{h}=h f(h / 2)+\int_{0}^{h}(&(x-h / 2) f^{\prime}(h / 2)+\frac{(x-h / 2)^{2}}{2} f^{\prime \prime}(h / 2) \\ &\left.+\frac{(x-h / 2)^{3}}{6} f^{\prime \prime \prime}(h / 2)+\frac{(x-h / 2)^{4}}{24} f^{\prime \prime \prime \prime}(h / 2)+\ldots\right) d x . \end{aligned} \nonumber$

Changing variables by letting $y=x-h / 2$ and $d y=d x$ , and simplifying the integral depending on whether the integrand is even or odd, we have

$\begin{aligned} I_{h}=& h f(h / 2) \\ &+\int_{-h / 2}^{h / 2}\left(y f^{\prime}(h / 2)+\frac{y^{2}}{2} f^{\prime \prime}(h / 2)+\frac{y^{3}}{6} f^{\prime \prime \prime}(h / 2)+\frac{y^{4}}{24} f^{\prime \prime \prime \prime}(h / 2)+\ldots\right) d y \\ =& h f(h / 2)+\int_{0}^{h / 2}\left(y^{2} f^{\prime \prime}(h / 2)+\frac{y^{4}}{12} f^{\prime \prime \prime \prime}(h / 2)+\ldots\right) d y \end{aligned} \nonumber$

The integrals that we need here are

$\int_{0}^{\frac{h}{2}} y^{2} d y=\frac{h^{3}}{24}, \quad \int_{0}^{\frac{h}{2}} y^{4} d y=\frac{h^{5}}{160} \nonumber$

Therefore,

$I_{h}=h f(h / 2)+\frac{h^{3}}{24} f^{\prime \prime}(h / 2)+\frac{h^{5}}{1920} f^{\prime \prime \prime \prime}(h / 2)+\ldots \nonumber$

6.1.2. Trapezoidal rule

From the Taylor series expansion of $f(x)$ about $x=h / 2$ , we have

$f(0)=f(h / 2)-\frac{h}{2} f^{\prime}(h / 2)+\frac{h^{2}}{8} f^{\prime \prime}(h / 2)-\frac{h^{3}}{48} f^{\prime \prime \prime}(h / 2)+\frac{h^{4}}{384} f^{\prime \prime \prime \prime}(h / 2)+\ldots, \nonumber$

and

$f(h)=f(h / 2)+\frac{h}{2} f^{\prime}(h / 2)+\frac{h^{2}}{8} f^{\prime \prime}(h / 2)+\frac{h^{3}}{48} f^{\prime \prime \prime}(h / 2)+\frac{h^{4}}{384} f^{\prime \prime \prime \prime}(h / 2)+\ldots \ldots \nonumber$

Adding and multiplying by $h / 2$ we obtain

$\frac{h}{2}(f(0)+f(h))=h f(h / 2)+\frac{h^{3}}{8} f^{\prime \prime}(h / 2)+\frac{h^{5}}{384} f^{\prime \prime \prime \prime}(h / 2)+\ldots . \nonumber$

We now substitute for the first term on the right-hand-side using the midpoint rule formula:

$\begin{array}{r} \frac{h}{2}(f(0)+f(h))=\left(I_{h}-\frac{h^{3}}{24} f^{\prime \prime}(h / 2)-\frac{h^{5}}{1920} f^{\prime \prime \prime \prime}(h / 2)\right) \\ +\frac{h^{3}}{8} f^{\prime \prime}(h / 2)+\frac{h^{5}}{384} f^{\prime \prime \prime \prime}(h / 2)+\ldots, \end{array} \nonumber$

and solving for $I_{h}$ , we find

$I_{h}=\frac{h}{2}(f(0)+f(h))-\frac{h^{3}}{12} f^{\prime \prime}(h / 2)-\frac{h^{5}}{480} f^{\prime \prime \prime \prime}(h / 2)+\ldots \nonumber$

6.1.3. Simpson’s rule

To obtain Simpson’s rule, we combine the midpoint and trapezoidal rule to eliminate the error term proportional to $h^{3}$ . Multiplying (6.3) by two and adding to (6.4), we obtain

$3 I_{h}=h\left(2 f(h / 2)+\frac{1}{2}(f(0)+f(h))\right)+h^{5}\left(\frac{2}{1920}-\frac{1}{480}\right) f^{\prime \prime \prime \prime}(h / 2)+\ldots, \nonumber$

$I_{h}=\frac{h}{6}(f(0)+4 f(h / 2)+f(h))-\frac{h^{5}}{2880} f^{\prime \prime \prime \prime}(h / 2)+\ldots . \nonumber$

Usually, Simpson’s rule is written by considering the three consecutive points $0, h$ and $2 h$ . Substituting $h \rightarrow 2 h$ , we obtain the standard result

$I_{2 h}=\frac{h}{3}(f(0)+4 f(h)+f(2 h))-\frac{h^{5}}{90} f^{\prime \prime \prime \prime}(h)+\ldots \nonumber$

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6.1.1. Midpoint rule

6.1.2. Trapezoidal rule

6.1.3. Simpson’s rule

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