Consider defined but non-integrable functions, y = f ( x ) .

Differentiation is straightforward, if often laborious.

The reverse process of integration is not often easy, or even
possible, except for well behaved forms, including most importantly,
parabolas.

Simpson method works by dividing the function into an even number of
intervals over the range of integration, computing f ( x ) at each end
of the intervals, fitting parabolas to every 3 points with common ends
to approximate f ( x ), and integrating the parabolas. Increasing the
number of intervals will yield arbitrarily greater precision and
accuracy. Parabolas are great for approximating f ( x ) since they
admit curvature.

Naturally I developed software to do the process but unfortunately,
f ( x ) has to be incorporated as a program statement in the code.
There may be some way to enter an arbitrary function during Visual
Basic v6 program execution, but I have yet to find it.

Note application of the method applied to :

y = Exp ( x / 2 ) * ( 100 - x * x )

https://postimg.cc/gallery/KZhYKvm

Note the method applied to :

y = Exp ( x / 2 ) * ( 100 - x * x )

https://postimg.cc/gallery/KZhYKvm