How is to find the value of $k$ for the following equation?
$$\int_{\{x:f(x)>k\}}f(x)\,dx=c,$$
where $c$ is a constant. Can anyone please show me a toy example from which I can understand how is to obtain the value of $k$?
2026-04-06 05:17:29.1775452649
$\int_{\{x:f(x)>k\}}f(x)dx=c$
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In general, this is a hard problem. Consider the graph below:
You're essentially asking to find the height of the horizontal line that makes the blue area equal to a given amount. Doing so will typically require:
Showing that the integral converges;
Finding a closed form for the integral;
Being able to invert that closed form to solve for $k$.
None of these are guaranteed for a general function. If you at least have convergence then you can use numeric methods to find an approximation, but even then it's likely to be difficult.