Hmm that's interesting, because it ininsinuates that the technique can't be done in mathematics...but it can, and it is the basis of calculus...limits.

So how did the "first mathemeticians", who didn't have the calculus formulas, figure out the area under the curve? Well, they did it just like the chemists. He divided the area under the curve into tall thin rectangles, of width x, and took the average value of the 2 y values at the top, got the area of the rectangles, and summed them up.

So then someone asked, "how can we get more accuracy?". They figured out as you narrowed the x width, the average y value at the top of the rectangle, "converged" to the same value. THE LIMIT.

It didn't take long for them to come up with the dx and dy notations, and to put it into formulas.

So even today, there is a whole branch of engineering called "applied numerical analysis", where you write programs exactly as described above. If your equation, or conditions, are so complex, that they can't be put into a simple formula, you break it up into intervals, either spatial or time, and compute the value of each interval separately, then sum up the results. Computers are so fast, it is usually the fastest way to model something.

I was in EE, loved the math, but dropped out to ponder the "meaning of it all" :-) Check out this story about how some people actually can see numbers as shapes....it makes you wonder about how puny our minds really are. Math Genius