Why approximate?

Integral calculus relies on antiderivatives: to evaluate $\displaystyle\int f$, we look for a function $F$ such that $F' = f$, then compute $F(b) - F(a)$. But an uncomfortable fact remains: many functions have no antiderivative that can be expressed using the usual functions.

The most famous example is the "bell curve" of probability:

$$f(x) = e^{-x^2}$$

This function has no antiderivative expressible in terms of the usual functions: it is impossible to write one with exponentials, logarithms or polynomials. Yet the area under this curve really does exist, and it is fundamental — the normal distribution, physics, statistics. So we have to compute it another way.

That is the role of numerical integration: instead of looking for an exact formula, we approximate the value of the integral by direct computation, from a few values of $f$.

Approach	What we compute
exact integral	$\displaystyle\int_a^b f = F(b) - F(a)$, if an antiderivative $F$ exists
numerical integration	$\displaystyle\int_a^b f \approx$ a combination of $f(x_0), f(x_1), \ldots, f(x_n)$

The idea builds on Riemann sums: split $[a \,;\, b]$ into $n$ pieces, evaluate $f$ at a few points, and combine these values. All the subtlety lies in the way of combining them: the cleverer it is, the better the approximation for the same number of points.

This is how calculators and computers work when you ask them for an "impossible" integral. Remember the starting point: when the antiderivative is missing, we compute the area numerically, with increasingly accurate methods.