The graphs of $y=x^4-2x^2+1$ and $y=1-x^2$ intersect at
three points. However, the area between the curves {can } be found
by a single integral. Explain why this is so, and write an integral
for this area.
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Tags: area, integration, higher-polynomial, quadratic, illustrative-example, a--g--v, t1, c2
The first function is the square of the second. A sketch of the graphs shows that the square is always greater than the second, so the signed area is the same as the unsigned area. Be sure students sketch this one, and encourage them to sketch whenever there’s any complexity. Even if particular students aren’t good sketchers, in a group, this is a chance for artistic students to contribute. In my experience, groups tend to settle on one or two students who are the “artists.”
– Eric Hsu 1998-08-09