Play with the Geogebra applet above to get a feeling for the relationship between , the height
of the upper boundary of the region at , and , the area of the region to the left of .
The example above is simple once you draw the picture! We'll see how this can be reduced to integrating . To make this visually obvious imagine that the region is composed of vertical tooth picks and then slide them so their butts all rest on the -axis. The tooth picks haven't changed length or thickness so the area hasn't changed either. (look up Cavalieri and his principle !)