# Mean Value Theorem for Integrals/Generalization

## 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:

- $\displaystyle \int_a^b \map f x \, \map g x \rd x = \map f k \int_a^b \map g x \rd x$

## Proof

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 15$: General Formulas involving Definite Integrals: $15.13$