= Weighted mean value theorem for integrals
{title2=$\int_a^b fg=f(\alpha)\int_a^b g$}
For real <continuous functions> $f,g$ on a compact interval with $g\geq0$, some $\alpha$ in the interval satisfies $\int fg=f(\alpha)\int g$. If $\int g>0$, the weighted average lies between the minimum and maximum of $f$, and the <intermediate value theorem> supplies $\alpha$. If $\int g=0$, the bound $|\int fg|\leq\sup|f|\int g$ makes both sides zero. Nonnegativity of the weight is essential to this argument.
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