# Differentation and integration

Differentiation and integration are (almost) inverse operations. Why is surface (integration) the opposite of slope (differentiation)?
In blue you see the graph of a certain function. The area under the curve is divided into n intervals. The area under the curve at each interval is approximated by a little rectangle shown in green. Subtract those green rectangular areas from the area under the blue curve. The little red shapes are the differences between the areas under the blue curve and the area of the corresponding green rectangles. Notice that the area of such a red shape is proportional to the slope of the curve at that interval. And the slope is just the derivative. So, the (graph of the) derivative of a function arises from subtracting area under the (graph of the) curve. In fact, so much area is subtracted, that there is no area left under two successive points. Now it is no surprise anymore that integration is the opposite operation, adding the area of such rectangles. The height of the first rectangle you add is free to choose. Thereafter, the little red shapes dictate the height of the rectangles, because all red shape must again line up to a single smooth curve. Conclusion: Integration connects with addition (of area) and differentation with subtraction (of area).