TITLE: What's the purpose of this formula?
QUESTION [5 upvotes]: 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)

REPLY [2 votes]: 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).