Editing Gaussian quadrature (section)

== Example of two-point Gauss quadrature rule ==
Use the two-point Gauss quadrature rule to approximate the distance in meters covered by a rocket from <math>t = 8\mathrm{s} </math> to <math>t = 30\mathrm{s},</math> as given by
<math display="block">s = \int_{8}^{30}{\left( 2000\ln\left[ \frac{140000}{140000 - 2100t} \right] - 9.8t \right){dt}}</math>

Change the limits so that one can use the weights and abscissae given in Table 1. Also, find the absolute relative true error. The true value is given as 11061.34 m.

Solution

First, changing the limits of integration from <math>\left[ 8,30 \right]</math> to <math>\left[ - 1,1 \right]</math> gives

<math display="block"> \begin{align}
\int_{8}^{30} {f(t) dt} &= \frac{30 - 8}{2} \int_{- 1}^{1}{f\left( \frac{30 - 8}{2}x + \frac{30 + 8}{2} \right){dx}} \\
&= 11\int_{- 1}^{1}{f\left( 11x + 19 \right){dx}}
\end{align} </math>

Next, get the weighting factors and function argument values from Table 1 for the two-point rule,

*<math>c_1 = 1.000000000 </math>
*<math>x_1 = - 0.577350269 </math>
*<math>c_2 = 1.000000000 </math>
*<math>x_2 = 0.577350269 </math>

Now we can use the Gauss quadrature formula
<math display="block"> \begin{align}
11\int_{-1}^{1}{f\left( 11x + 19 \right){dx}} & \approx 11\left[ c_1 f\left( 11 x_1 + 19 \right) + c_2 f\left( 11 x_2 + 19 \right) \right] \\
&= 11\left[ f\left( 11( - 0.5773503) + 19 \right) + f\left( 11(0.5773503) + 19 \right) \right] \\
&= 11\left[ f(12.64915) + f(25.35085) \right] \\
&= 11\left[ (296.8317) + (708.4811) \right] \\
&= 11058.44
\end{align}</math>
since
<math display="block"> \begin{align}
f(12.64915) & = 2000\ln\left[ \frac{140000}{140000 - 2100(12.64915)} \right] - 9.8(12.64915) \\
&= 296.8317
\end{align}</math>
<math display="block"> \begin{align}
f(25.35085) & = 2000\ln\left[ \frac{140000}{140000 - 2100(25.35085)} \right] - 9.8(25.35085) \\
&= 708.4811
\end{align}</math>

Given that the true value is 11061.34 m, the absolute relative true error, <math>\left| \varepsilon_{t} \right|</math> is
<math display="block"> \left| \varepsilon_{t} \right| = \left| \frac{11061.34 - 11058.44}{11061.34} \right| \times 100\% = 0.0262\% </math>