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11

# Integral Inequalities

## 11.11 Mean Value Theorems

### 11.111 First mean value theorem

Let f(x) and g(x) be two bounded functions integrable in [a, b] and let g(x) be of one sign in this interval. Then

${\int }_{a}^{b}f\left(x\right)g\left(x\right)\text{d}x=f\left(\xi \right){\int }_{a}^{b}g\left(x\right)\text{d}x,$

CA 105

with a ≤ ξ ≤ b.

### 11.112 Second mean value theorem

(i)  Let f(x) be a bounded, monotonic decreasing, and nonnegative function in [a, b], and let g(x) be a bounded integrable function. Then,

${\int }_{a}^{b}f\left(x\right)g\left(x\right)\text{d}x=f\left(a\right){\int }_{a}^{\xi }g\left(x\right)\text{d}x,$

with a ≤ ξ ≤ b.

(ii)  Let f(x) be a bounded, monotonic increasing, and nonnegative function ...

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