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

\[\int_{a}^{b}f(x)dx=f(\xi)\cdot\int_{a}^{b}1dx=f(\xi)(b-a).\]

| | | | | created: 2016-03-06 20:11:33 | modified: 2016-03-06 20:18:03 | by: *bookofproofs* | references: [581]

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[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen”, Vieweg Studium, 1983