Numerical Analysis/Romberg Excercise

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Use Romberg Integration to compute $R_{3,3}$ for $\int_0^1 x^2 e^{-x} \,dx$

Solution:

$R_{1,1} = \frac{h_{1}}{2}[f(0)+f(1)]$
$R_{1,1} = \frac{1}{2}[0+\frac{1}{e}]$
$R_{1,1} = .1839397206$

$R_{2,1} = \left(\frac{1}{2}\right)[R_{1,1}+h_{1}f(a+h_{2})]$
$R_{2,1} = .1379547904$

$R_{3,1} = \left(\frac{1}{2}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))]$
$R_{3,1} = .1475727039$

$R_{2,2} = R_{2,1}+ \frac{R_{2,1}-R_{1,1}}{4-1}$
$R_{2,2} = .1226264803$

$R_{3,2} = R_{3,1} + \frac{R_{3,1}-R_{2,1}}{4-1}$
$R_{3,2} = .1507786751$

$R_{3,3} = R_{3,2} + \frac{R_{3,2}-R_{2,2}}{16-1}$
$R_{3,3} = .1526554881$

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