In Germany, one usual example to start teaching about integrals is to look at a simple (piecewise constant or with constant slope) functions that make up a water flow vs. time diagram and ask about the total flow during a given time span. This leads to the idea that the area under the curve is of interest and represents exactly the desired quantity.
I was thinking about a discretisation of this example because my dog requires pain medication and I plotted a daily dose vs. day graph. This is basically the same as the water flow diagram, but already with discrete steps, probably leading to the "rectangular stripes" method.
Is there any research or experience about this approach?