︠b4b393a2-3056-4291-bb09-53e1a73fdb49i︠ %html

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\n

Play 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\\).

"} ︠f215a5c1-3507-45ae-abf4-8e709aecfed1︠ ︡d9923fb5-3e37-44a4-882e-c21ca42ee671︡ ︠0f2cc388-67ce-4059-affc-530b58e92de5︠ ︡3eb54264-8664-4282-a225-3ebb3281d2a4︡ ︠4bd0cd7d-0507-404b-addc-fcc06b71f86f︠ ︡b61375d2-92a0-476a-aaf2-016433be53df︡ ︠e7ce8ced-ab82-45b6-809f-91b424bbacf8︠ ︡a00e6595-7b39-4f0c-93f3-43dbba7865bd︡ ︠61beb1f4-4b3f-4af3-af9e-dedbb2228293︠ ︡98b32ecd-8b46-429e-9dc4-e277134af380︡ ︠16a756ae-6c39-4610-a0a3-3a29e6ca8878︠ ︡146dce1e-bc39-4dc1-bc4f-d49bd24416c3︡ ︠70af2aac-ce7b-4d2d-b8db-08780d2ce4a7︠ ︡bcba13e2-f344-417b-8643-d94883009d40︡ ︠674ba8fa-9afd-424a-8364-d4cd322092e8︠ ︡b79fe664-53f7-4f86-91c7-780e25fa8431︡ ︠810ddad2-e11d-46d2-b54f-8a16be572bb5︠ ︡d0908c3f-f89c-4425-b314-7871c1560457︡ ︠72884d21-f3fb-41f2-b832-5d8517a0a9a6︠ ︡a01ddc4c-c93a-4bdc-b0a0-ae70bd13285a︡ ︠0145ca48-707e-4ff7-9297-7c4d075fe42e︠ ︡f1ccd155-272a-4b53-8b8d-474a1384518b︡ ︠6fa19698-b3fe-48a2-b61c-9483d771620a︠ ︡5a763af2-c24a-4feb-83ff-82957f3cf08d︡ ︠3e7c6ec2-a783-494c-85e6-74c302977f6a︠ ︡43345ff3-e011-4d9a-b220-d393bdd038f7︡ ︠d7f1030d-b881-4d90-9387-584032cfb581︠ ︡8f8b9d7e-0a00-43e2-843d-d69f539acbf9︡ ︠9ea15c0e-91e5-44e2-bf43-95ec65bcf0eei︠ %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 !)"} ︠60dcda98-f530-45c8-b304-d1f078374146︠