Continuous functions attain their mean value.

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Now we present the mean value theorem of integration and its manifold
consequences.

**Theorem 1** (Mean Value Theorem of Integration).
*Let \(f,g:[a,b]\to\mathbb{R}\) be
continuous and let \(g(x)\geq0\) for
all \(x\in[a,b]\) (i.e., \(g\geq0\)). Then there exists some \(\hat{x}\in[a,b]\) such that \[\int_a^bf(x)g(x)\, dx=f(\hat{x})\cdot
\int_a^bg(x)\, dx.\]*

*Proof:* Let \[m=\min\{f(x)\::\:x\in[a,b]\},\qquad
M=\max\{f(x)\::\:x\in[a,b]\}.\] Since \(g\geq0\), for all \(x\in[a,b]\) holds \(mg(x)\leq f(x)g(x)\leq Mg(x)\) and, by the
monotonicity of the integral holds \[m\int_a^bg(x)\, dx=\int_a^bmg(x)\, dx\leq
\int_a^bf(x)g(x)\, dx\leq \int_a^bMg(x)\, dx\leq M\int_a^bg(x)\,
dx.\] Then there exists some \(m\leq\mu\leq M\) such that \[\mu\int_a^bg(x)\, dx=\int_a^bf(x)g(x)\,
dx\] Since \[\min\{f(x)\::\:x\in[a,b]\}\leq\mu\leq\max\{f(x)\::\:x\in[a,b]\},\]
the intermediate value theorem implies that there exists some \(\hat{x}\in[a,b]\) with \(\mu=f(\hat{x})\) and thus \[f(\hat{x})\int_a^bg(x)\, dx=\int_a^bf(x)g(x)\,
dx.\] \(\Box\)

Let \(f:[a,b]\to\mathbb{R}\) be
continuous. Then there exists some \(\hat{x}\in[a,b]\) such that \[\int_a^bf(x)\, dx=f(\hat{x})\cdot
(b-a).\]

*Proof:* Apply the mean value theorem to \(g=1\). Observing that \[\int_a^bg(x)\, dx=\int_a^b1\, dx= b-a,\]
the result then follows immediately. \(\Box\)

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