Now that was interesting, you say. But how about a little

*proof*? For a mathematician, the word "proof" is like waving the red handkerchief to a bull at Pamplona, but I'm not going there. However, it is always a good idea to check things a bit to see if they make sense. Consider the simple equation:

We want to know the area under the curve.

I know how to do that, you say. Remembering that the curve itself is the derivative of the area function f(x), we have:

That was way too easy. But consider this: how about the area

*above*the curve?

Notice that we have

I get a little freaked out seeing x as a function of y, so I am going to mentally turn the curve through 90 degrees and then make its mirror image reflection. I get this:

Now, our differentiation trick works for fractional exponents. So we have:

And, as we might hope and expect, 2/3 + 1/3 does equal the area of the unit square. And note that it will work for any power of x. If, for example, we start with the curve y = x

^{3}, then the first area is 1/4 x

^{4}, and the second area is 3/4 x

^{4/3}. It's easy to see that it works for any rational power of x. Suppose we have:

As we guessed, we see that b/c + a/d = 1.

(Of course, it helps that we chose the upper limit for measuring the area at x = 1, y = 1. Evaluation for some other bound would involve a bit more calculation).

Ladies and gentlemen, will you please…give it up for Mr. Leibnitz and Mr. Newton.

[Update: blogger messed with my code. Think I fixed it:]

R code: