let dx be just a number. troughout the function f you with an interval D measured y=f(x) for x, and sum upp the rectangles with side dx and height y=f(x), this is called the undersum, an approximation of the area below the function, (negative area if below 0) now similarly you can start at y=f(D+x) and get the oversum, which are rectangles above the function when it raises. Now if you decrease D the areas will get more accurate, and when dx is infinitely small, you can prove that there's exactly one value I for the following equation
S1(D)>= I >= S2(D)
where S1 and S2 are the over respective under sums of the rectangles with width D and interval D, when D is an infinitely small number. We call this dx, and I we call the intergral for f.