Numerical Analysis/Romberg Excercise

Use Romberg Integration to compute $R_{3,3}$ for $\underset{0}{\overset{1}{\int }}{x}^2{e}^{-x}dx$

Solution:

$R_{1,1}=\frac{h_1}{2}[f(0)+f(1)]$
$R_{1,1}=\frac{1}{2}[0+\frac{1}{e}]$
$R_{1,1}=.1839397206$

$R_{2,1}=\left(\frac{1}{2}\right)[R_{1,1}+h_{1}f(a+h_2)]$
$R_{2,1}=.1379547904$

$R_{3,1}=\left(\frac{1}{2}\right)[R_{2,1}+h_2(f(a+h_3)+f(a+3h_3))]$
$R_{3,1}=.1475727039$

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

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

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