Numerical Analysis/Romberg Example

Use Romberg Integration to compute $R_{3,3}$ for the following integral
$\underset{0}{\overset{\frac{\pi }{2}}{\int }}cosxdx$

Solution:

$R_{1,1}=\frac{\pi }{4}[cos(0)+cos(\frac{\pi }{2})]$

$R_{1,1}=\frac{\pi }{4}$

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

$R_{2,1}=\left(\frac{1}{2}\right)[\frac{\pi }{4}+\frac{\pi }{2}cos\left(\frac{\pi }{4}\right)]$
$R_{2,1}=1.178023457$

$R_{3,1}=\left(\frac{1}{2}\right)[R_{2,1}+h_2(f(a+h_3)+f(a+3h_3))]$
$R_{3,1}=\left(\frac{1}{2}\right)[1.178023457+\frac{\pi }{4}(cos(\frac{\pi }{8})+cos(\frac{3\pi }{8})]$
$R_{3,1}=1.374317658$

$R_{2,2}=R_{2,1}+\frac{R_{2,1}-R_{1,1}}{4-1}$
$R_{2,2}=1.178023457+\frac{.3926252936}{3}$
$R_{2,2}=1.308898555$

$R_{3,2}=R_{3,1}+\frac{R_{3,1}-R_{2,1}}{4-1}$
$R_{3,2}=1.374317658+\frac{.196294201}{3}$
$R_{3,2}=1.439749058$

$R_{3,3}=R_{3,2}+\frac{R_{3,2}-R_{2,2}}{16-1}$
$R_{3,3}=1.439749058+\frac{.1308505033}{15}$
$R_{3,3}=1.448472425$