CoCalc Public Files2017-02-09-112949.sagews
Author: [email protected] Gage
Views : 40
Description: embedded webwork problem
Compute Environment: Ubuntu 18.04 (Deprecated)

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 !)