TITLE: What is the difference between a Summation and an Integration?
QUESTION [5 upvotes]: What is the difference between a Summation and an Integration?
Both of them add some values. Right? Then what is the difference?
Please, explain in layman's terms.

REPLY [6 votes]: Summation uses "discrete" values (1, 2, 3, 4...), while integration usually uses continuous values over an uncountably infinite interval (0 to infinity, for example, or even 0 to 1). That is, on a number line, $\sum$ summation skips quite a few values!
Kind of cool note: this difference is really what kick-started quantum mechanics. In observing the graphs of intensity vs wavelength at specific temperatures, there were repeated failed attempts at modelling an equation  which would recreate the behavior of the graph.
All of the attempts, which used integration, would perform decently at one end but would diverge to infinity, until Planck made the radical assumption at the spectrum was built on discrete values, not continuous, and switched the integrations to sums, which then perfectly modelled the observations.