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Normally, we have a function $f(x)$ which we integrate in order to find the area under the curve from some a to some b....

but is it possible to solve an equation going in the other direction? Meaning that if I know the area under a given curve $f(x)$ from $x = 2$ to $x = 8$ is $4$, I can then calculate $f(x)$, or all the $f(x)$s that would satisfy my equation?

usermath
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Dann
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1 Answers1

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In general no, imagine you know the area under the graph from $x=2$ to $x=8$ is $4$, there are an infinite amount of continuous curves satisfying this.

A way to narrow things down, is to consider integrals of a different nature, I.e. find $f(x)$ such that $f$ is either a minimum or maximum of $\int_a^b\sqrt{f'(x)^2+f(x)^2}dx$

Ellya
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  • just a tiny remark: don't take the maximum here :-) since $L^{1}$ is no subspace of $L^{2}$ – Max May 12 '14 at 18:14
  • Was just an example :), it is meant to be similar to the area functional I think. – Ellya May 12 '14 at 18:17