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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 \(x\).
︡ce2d99a6-a563-4aeb-88ae-110f39332cdb︡{"done":true,"html":"\n\nPlay with the Geogebra applet above to get a feeling for the relationship between \\(f\\), the height
\nof the upper boundary of the region at \\(x\\), and \\(F\\), the area of the region to the left of \\(x\\).
"}
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%md
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](https://en.wikipedia.org/wiki/Bonaventura_Cavalieri) and his principle !)
︡3a8941f4-6d6c-4eeb-86c6-dbcba1ca040a︡{"done":true,"md":"\n\nThe example above is simple once you draw the picture! We'll see how\nthis can be reduced to integrating $f(x)-g(x)$. To make this visually\nobvious imagine that the region is composed of vertical tooth picks and\nthen slide them so their butts all rest on the $x$-axis. The tooth picks\nhaven't changed length or thickness so the area hasn't changed either.\n(look up [Cavalieri](https://en.wikipedia.org/wiki/Bonaventura_Cavalieri) and his principle !)"}
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