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## Theorem: Mean Value Theorem For Riemann Integrals

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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