Why Can’t You Just Integrate e^{-x^2}, Anyway?


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

y = e^{-x^2}

What we mean by that is the area between some left boundary, x = a , and some right boundary, x = b , 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 (b - a)^2 e^{-(b - a)^2} or something normal like that. The one interesting exception is that you can find the area if the left bound is -\infty and the right bound +\infty . 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.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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