A detailed solution without AI is required Problem $2$: A New Approximation

The species of MathLand $($some of which descended to Earth to conduct MathLab$)$ are far too evolved, they believe that straight lines are elementary and outdated! They prefer neater splines to make approximations. Unfortunately they don't have any computational wisdom and are stuck! They need an algorithm that will help them integrate a function. Can you be their saviour and help them write the code to do so$?$

Hint: Be crcative and try something new or Think Simpsons!

If you're developing your own method write out all the approximations and assumptions clearly, Remember: Taylor is always your friend and the approach always carries more importance than the final answer.

Final Expected Submission: A script written in MATLAB or GNU Octave that implements the following:

Your own new algorithm for approximating integrals OR$.$ Simpson's method for calculating a definite integral.

Tests out your algorithm on some integral $($for example ${\displaystyle \underset{0}{\overset{2}{}}}{x}^{2}dx$ or any integral of your choice for that matter$)$ and see if your integrator is working.