Mean Value Theorem for Integrals/Generalization

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Theorem

Let $f$ and $g$ be continuous real functions on the closed interval $\closedint a b$ such that:

$\forall x \in \closedint a b: \map g x \ge 0$

Then there exists a real number $k \in \closedint a b$ such that:

$\ds \int_a^b \map f x \map g x \rd x = \map f k \int_a^b \map g x \rd x$


Proof

Let:

$\ds \int_a^b \map g x \rd x = 0$

We are given that:

$\forall x \in \closedint a b: \map g x \ge 0$

Hence by Continuous Non-Negative Real Function with Zero Integral is Zero Function:

$\forall x \in \closedint a b: \map g x = 0$


Hence:

$\ds \int_a^b \map f x \cdot 0 \rd x = \map f k \cdot 0$

and so the result holds for any choice of $k$.

$\Box$


Let:

$\ds \int_a^b \map g x \rd x \ne 0$

From Continuous Real Function is Darboux Integrable, $f$ is Darboux integrable on $\closedint a b$.

By the Extreme Value Theorem, there exist $m, M \in \closedint a b$ such that:

$\ds \map f m = \min_{x \mathop \in \closedint a b} \map f x$
$\ds \map f M = \max_{x \mathop \in \closedint a b} \map f x$

Then the following inequality holds for all $x$ in $\closedint a b$:

$\map f m \le \map f x \le \map f M$


Multiplying by $\map g x$, and using that:

$\forall x \in \closedint a b: \map g x \ge 0$

we get:

$\map f m \map g x \le \map f x \map g x \le \map f M \map g x$

Integrating from $a$ to $b$ gives:

$\ds \int_a^b \map f m \map g x \rd x \le \int_a^b \map f x \map g x \rd x \le \int_a^b \map f M \map g x \rd x$

By Linear Combination of Definite Integrals:

$\ds \map f m \int_a^b \map g x \rd x \le \int_a^b \map f x \map g x \rd x \le \map f M \int_a^b \map g x \rd x$

Dividing by $\ds \int_a^b \map g x \rd x$ gives:

$\ds \map f m \le \dfrac {\int_a^b \map f x \map g x \rd x} {\int_a^b \map g x \rd x} \le \map f M $

By the Intermediate Value Theorem, there exists some $k \in \openint a b$ such that:

\(\ds \dfrac {\int_a^b \map f x \map g x \rd x} {\int_a^b \map g x \rd x}\) \(=\) \(\ds \map f k\)
\(\ds \leadsto \ \ \) \(\ds \int_a^b \map f x \map g x \rd x\) \(=\) \(\ds \map f k \int_a^b \map g x \rd x\)

$\blacksquare$


Sources

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