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Pattern – 30 : Definite integration

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⋆ Pattern – 30 :

If m is the least value (global minimum) and M is greatest value (global maximum) of f(x) for ∀ x ∈ [a, b]

i.e. m ≤ f(x) ≤ M ; x [a, b],

then prove that

m (b – a) ≤ $\int_{a}^{b}f\left(x\right)\ dx$ ≤ M (b – a)

Proof :

∀ x ∈ [a, b]

m ≤ f(x) ≤ M

$\therefore\ \ \int_{a}^{b}{m\ dx}\le\ \int_{a}^{b}f\left(x\right)dx\le\ \int_{a}^{b}M\ dx$

$\therefore\ \ m\left(b-a\right)\le\ \int_{a}^{b}f\left(x\right)dx\le\ M\left(b-a\right)$

$\left(\because\ \int_{a}^{b}k\ dx=k\left[x\right]_a^b=k\left(b-a\right)\right)$