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Pattern – 29 : Definite integration

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⋆ Pattern – 29

f is continuous on [a, b] and | f(x) | ≤ k, ∀ x ∈ [a, b], then $\left|\int_{a}^{b}f\left(x\right)dx\right|$ ≤ k (b-a)

Proof :

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

$\le\ \int_{a}^{b}kdx$

$=k\left[x\right]_a^b$

= k(b-a)

Thus,

$\left |\int_{a}^{b}f\left(x\right)dx\right|\le k\left(b-a\right)$