Let \([a,b]\) be a closed real interval and let \(f,\phi:[a,b]\mapsto\mathbb R\) be continuous functions with \(\phi(x)\ge 0\) for all \(x\in[a,b]\). Then there exists a value mean value) \(\xi\in[a,b]\) such that
\[\int_{a}^{b}f(x)\phi(x)dx=f(\xi)\cdot\int_{a}^{b}\phi(x)dx.\]
For the special case \(\phi(x)=1\) for all \(x\in[a,b]\), we have
\[\int_{a}^{b}f(x)dx=f(\xi)\cdot\int_{a}^{b}1dx=f(\xi)(b-a).\]
Proofs: 1
