What's the purpose of this formula?

Question

Just found this image on the web:

Can anyone explain what's the meaning (if any) of this formula?

(I did a Google image search but found no answer)

Answer 1

It's the natural approach to calculate an integral :
Splitting your segment in tiny pieces so that you can make the approximation that the value of your function on each of these segments is constant.

Note that this approach has a lot of applications : I have OFDM in mind but there are a lot more.

The following gif came from here

Answer 2

The definite integral of $f(x)$ from $a$ to $b$ is defined as the limit as the size of the changes in $x$ go to zero of the sum of the size of the changes in $x$ times the value of $f(x)$ after each change for every $x$ after each change. Don't know what the $n$ is doing in there, probably signifies $b-a \over \Delta_i x$.

Answer 3

You can type on google Riemann integral and summation. This link could be useful to you Riemann Sums and definite integrals.

Answer 4

Say you have a continuous function $f$, you plot it and want to calculate the area under the resulting curve over an interval $\langle a,b \rangle$. You would probably measure the value of the function at specific points of the interval and multiply it with the distance between those points, to get approximation of the actual area under the curve by calculating the area of rectangles enclosed under the curve. The more such rectangles you make, the more precise the value for the area you get.

This formula says the definite integral from $a$ to $b$, which is possible to interpret as the area under $f$ on the interval from $a$ to $b$ (left hand side) is equal to summing the areas of more and more rectangles under $f$ (right hand side).