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Pattern – 28 : Definite integration

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⋆ Pattern – 28

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

Proof :

We know that,

– | f(x) | ≤ f(x) ≤ | f(x) |

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

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

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