What is integration in math and why is it important?

[challa]

I give a smart kindergartner a rectangle with height 2m and width 4m; I ask the kid to find the area. The kid is smart and tells me 2x4=8m.

I give the kid a drawing of a God-awful figure with a curves on top with various heights. I ask for the area. How would you find it? You could draw smaller rectangles under and over the the curves and have an approximate figure made from rectangles, and you can find the area of each rectangle and add them up; that'll give you a good estimate of the area. http://upload.wikimedia.org/wikipedia/commons/2/2c/Integral_Riemann_sum.png

But that's not going to be exact, so you draw smaller, thinner rectangles. That gives you an area closer to the precise number, but that's still not exact. You could make infinitely smaller rectangles and you will only come that much closer to the exact number.

So you take the integral and every is now okay.