Use Calculus results (not numeric integration methods) to approximate definite integral

Consider$$\int_{-1}^2 e^{-x^2}dx$$

How can I approximate its value using Calculus theory? That is, not using numerical methods such as the trapezoid rule taught in Numerical Mathematics courses?

Ideas:

1. Riemann sums (but seems too coarse).
2. Use the fact the integral of the function over all the reals equals $\sqrt{\pi}$, but I am not sure how to use it.
3. Mean Value Theorem, but again I am not sure how to use it (how to choose the point).
4. Taylor series.

Any ideas?