Research the Romberg method to approximate the integral of a given function. The research should cover the

following points:

A brief introduction of the method $($covers who created it$,$ when? Why, and how to apply it$).$

The advantages and disadvantages of the method compared to the method we studied

$($Trapezoidal and Simpson's$).$

Let ${\displaystyle \underset{0}{\overset{1}{}}}x{e}^{x}dx$ and $h=0\mathrm{.}2$

a$)$ Use Simpson's method to approximate the integral.

b$)$ Apply Romberg integration to approximate the integral $($Stop at $R_{33})$

c$)$ Which method gives a better approximation? Is this what you expected? Justify your answer.

List of references $($at least $2)$ written in Harvard referencing style. Note that Wikipedia is not

acceptable as a reliable academic resource.