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Pattern – 27 : Definite integration

March 1, 2021
Category: Pattern – 27 : Definite integration

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⋆ Pattern – 27

∀ x ∈ [ a, b ], g(x) ≤ f(x) ≤ h(x), then $\int_{a}^{b}g\left(x\right) dx$ ≤ $\int_{a}^{b}h\left(x\right) dx$ (where, a<b) deduce that, g(x) ≥ 0, then $\int_{a}^{b}g\left(x\right) dx$ ≥ 0

Proof :

$\int_{a}^{b}f\left(x\right)dx$

Obviously,

$\int_{a}^{b}g\left(x\right)dx\le\ \int_{a}^{b}f\left(x\right)dx\le\ \int_{a}^{b}h\left(x\right)dx$

g(x) ≥ 0 ⇒ 0 ≤ g(x)

$\ \Rightarrow\ \int_{a}^{b}0\ dx\le\ \ \int_{a}^{b}g\left(x\right)dx$

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Pattern – 27 : Definite integration

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