You needed to submit a pdf document where you discuss optimization techniques to the following numerical methods provided below. One way to support your claim is to benchmark your approach and check the time it takes to perform the same task with different techniques. The reason why this can be helpful is that a lot of these High-Performance approaches are dependent on a particular system's architecture and this includes possible unforeseen dependencies that can influence the execution time. You will not get a good score If you do not turn in a document (preferably in PDF) without arguing why your particular approach works. You are a college student who is about to graduate so, please submit work that is representative of you.
Ps- Keep in mind that sometimes, a serial approach may be faster than a parallel approach and some instances will require that you modify an algorithm to make it parallel. You are free to optimize your work however you see fit.
Code: https://rosettacode.org/wiki/Numerical_integration#C
Wiki: https://en.wikipedia.org/wiki/Numerical_integration
Code: https://www.geeksforgeeks.org/runge-kutta-2nd-order-method-to-solve-differential-equations/
Wiki: https://en.wikipedia.org/wiki/Runge–Kutta_methods
For project ; you will likely need to write a document in LaTex as a word document will not work sufficiently. I personally would recommend overleaf.com for doing so. If you didn't get an invite to the site previously, either ask one of your classmates to invite you or simply sign up there. For writing equations, I would advise using an equation editor until you get used to the LaTex syntax.
Link to equation editor: http://www.codecogs.com/latex/eqneditor.php
Link to LaTex template: https://www.overleaf.com/read/gkpkqyfqrdgc
Like Galileo's scientific method, you want to satisfy your assumption with empirical data, which are the benchmarking.
What you need to do for this project is try multiple approaches and see how they compare to one another. You then have to bench mark the said method on large parameters since other processes can influence the execution time greatly if the size relatively small.