What is the difference between a Summation and an Integration? 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.
 A: 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.
A: 
What is the difference between summation and integration ? Please, explain in layman's terms.

Take a look at this picture. Summation is the part that looks like a city skyline, and integration is the portion that resembles a mountain outline.
A: The integration is, in some sense, the sum on an uncountable number of elements. For example, if you take a integer $n$, there exists a smaller closer integer $n-1$. But if you take a real $x$, there doesn't exist any smaller closer real number, it is why we define a infinitesimal variation $\text{d}x$ in integration.
A sum is an integration on a countable set, e.g. $\mathbb{N}$ where the variation $\text{d}x=1$.
A: A summation applies  to finite, countable sets ie: integers, rational numbers etc. Conversely, integration occurs  over discrete or infinite bounds, but more importantly  over the reals, which is not a countably finite set.
