Connection between summation and integration

[Jhenrique]

Hellow!

I want you note this similarity:
$$\\ \int xdx=\frac{1}{2}x^2+C \\ \int x^2dx=\frac{1}{3}x^3+C$$
$$\\ \sum x\Delta x=\frac{1}{2}x^2-\frac{1}{2}x+C \\ \\ \sum x^2\Delta x=\frac{1}{3}x^3-\frac{1}{2}x^2+\frac{1}{6}x+C$$

Seems there be a connection between the discrete calculus and the continuous. Exist some formula that make this connection? Given the summation of a function f(x) is possible to know the integral of f(x), or, given the integral of a function f(x) is possible know the summation of f(x)?

 

[pwsnafu]

This is studied in time scales calculus, a very active field of research today. It's only been around for thirty or so years