Computational complexity of numerical integration of gaussian function

Question

$ \int^{b}_{a} \exp(-x^2)\,dx$.

I have the following two questions regarding the above integral expression of the Gaussian function:

1. Is there a numerical method we can use to solve the above integral? It doesn't have to be the most efficient method.

2. Would there be a way to determine the computational complexity of such a numerical method?

Answer 1

I suppose you mean $$\int^{b}_{a} \exp(-x^2)\,dx=\frac{\sqrt{\pi }}{2} \Big(\text{erf}(b)-\text{erf}(a)\Big)$$ This comes from the definition of the ${erf}$ function.

At http://en.wikipedia.org/wiki/Error_function, you will find approximation methods of different complexities.