John Quintanilla, at the Mean Green Math blog, started a set of essays about numerical integration. And the good questions, too, like, why do numerical integration? In a post fram last week he looks at one integral that you encounter in freshman calculus, to learn you can’t do except numerically. (With a special exception.) That one is about the error function, also called the bell curve. The problem is finding the area underneath the curve described by
What we mean by that is the area between some left boundary, , and some right boundary, , that’s above the x-axis, and below that curve. And there’s just no finding a, you know, answer. Something that looks like (to make up an answer) the area is or something normal like that. The one interesting exception is that you can find the area if the left bound is and the right bound . That’s done by some clever reasoning and changes of variables which is why we see that and only that in freshman calculus. (Oh, and as a side effect we can get the integral between 0 and infinity, because that has to be half of that.)
Anyway, Quintanilla includes a nice bit along the way, that I don’t remember from my freshman calculus, pointing out why we can’t come up with a nice simple formula like that. It’s a loose argument, showing what would happen if we suppose there is a way to integrate this using normal functions and showing we get a contradiction. A proper proof is much harder and fussier, but this is likely enough to convince someone who understands a bit of calculus and a bit of Taylor series.