Play with the Geogebra applet above to get a feeling for the relationship between $f$, the height

of the upper boundary of the region at $x$, and $F$, the area of the region to the left of $x$.

The example above is simple once you draw the picture! We'll see how this can be reduced to integrating $f(x)-g(x)$. 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 $x$-axis. The tooth picks haven't changed length or thickness so the area hasn't changed either. (look up Cavalieri and his principle !)