Well let me try in English but in non-calculus terms.

Consider a sheet of graph paper and draw your self an origin and an x & y coordinate system. Now draw an arbitrary line lets say slanting up to the right at about 45 degrees. Make that as long as you like then drop a perpendicular to the x axis from its end.

So now you have a triangle and what we need- for whatever reason- is its area. The easiest way since you've got graph paper is to count the squares. Think of it as lots of columns- each column is as wide as the small squares on your paper, perhaps 1mm or maybe 1/10", whatever. So you can say that each column is that wide, and as high as the slanting line at that x-distance from the origin.

Since the height of the column is a function of the equation of the line (in this case y=mx+c 'cos it's a straight line) you can calculate the height as opposed to having to measure it, and the area is the sum of the heights of all columns times the width.

In that case, with a discrete width (eg the 1mm or whatever) we would consider the area to be the capital sigma. We'd call the column width delta-x.

Problem is , if you draw this, is that at the top of the column you've got a triangle and we're actually not 100% accurate 'cos we've counted as area, a small piece actually outside.

So make the column infinitesimal ; then our delta-x is written as dx and the exact height at any x value, times dx gives us the exact area of that skinny column. When we add them up, we get the exact area, and that's when we use the elongated 's' which you failed so misreably to draw earlier.