# 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?

• Summation iterates among a discrete set (you can sum all the elements in the set $\left\{1,2,...,10 \right\}$ but not the element of the set $\left[1,10 \right]$ (real closed interval for example)). For the second one the integration is defined. In laymans terms i think it's the structure of the domain the difference. – user8469759 Nov 2 '15 at 11:39
• The real question is: what are the similarities? – Jack M Nov 2 '15 at 13:42
• @JackM OP already noted the largest similarity in the question - "both of them are sums" – galois Nov 9 '15 at 5:19

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.

• Really f'ing cool note – user Feb 23 at 6:56

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.

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 $\text{d}x=1$.

• Summation adds discrete terms while integration means adding by parts it is also used for finding area under the curve . For this we cannot ise summation. – Archis Welankar Nov 2 '15 at 11:46
• @MoebiusCorzer Can you elaborate in the infinitesmall variation dx in integration? In general: $$\sum_{x=i}^{n}x$$ isn't the same operation as $$\int_{a}^{b} xdx$$ – user599310 Mar 17 '20 at 17:40

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.