Can I do an integral, possibly using gaussian quadrature, when the abscissas are fixed (for reasons that I don't want to get into right now), i.e. is it possible to calculate the weights for fixed abscissas that I don't get to choose?



do numerical integration with fixed abscissas
My interpretation of the problem: given $n$ pairs $(x\_j, f(x\_j)$ with 
$a \le x\_1 < x\_2 < \ldots < x\_n \le b$, you want an approximation 
to $\int\_a^b f(x)\, dx$. 


One way, that would give the correct value for polynomials
of degree $\le n-1$, would be to use $\int\_a^b g(x)\, dx$ where $g$ is the Lagrange interpolating polynomial of degree $\le n-1$ corresponding to your data. However, that
can be very unstable (see "Runge's phenonenon"). A better idea might be to use something like a composite quadrature rule: partition your interval into subintervals, each containing a few of the $x\_j$, and use the interpolation for those $x\_j$ in the subinterval to approximate the integral over that subinterval. 


