I am not familiar with Nyquist-Shannon sampling theorem. Do you mean you want to calculate some data based on the formula described in
http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem? It seems to use Fourier transform. As far as calculate an approximate value for a definite integral programmatically, you can use the definition of definite integral:
Divide the region to an infinite number of subintervals, calculate each subinterval’s area, and add them together. One of the solutions is to use a rectangle to emulate each subinterval. Of course we won’t be able to divide the region infinitely.
So we can approximate the result by, say use 1000 intervals. It’s easy for a computer to calculate the areas for 1000 rectangles. We can also use Simpson’s Rule which may give us better result (use quadratics instead of rectangles). I am not sure
how to calculate Fourier transform as it seems to require calculate a definite integral from negative infinite to positive infinite. But according to
http://en.wikipedia.org/wiki/Fourier_transform, in most cases you can use –T/2 and T/2 as the lower/upper bound, so this is essentially a definite integral. And since Fourier
transform seems to assume the original function converges between the interval, I “guess” this approximation can work in most cases.
In addition, if you have more questions regarding math, I would recommend you to post a thread on a math forum, there are more related experts in math forums.