Evaluate the following integrals: ^4_0|x-1|dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^4_0|\text{x}-1|\text{dx}$

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Answer

$\int\limits^4_0|\text{x}-1|\text{dx}$
We know that,
$|\text{x}-1|=\begin{cases}-(\text{x}-1),&0\leq\text{x}\leq1\\\text{x}-1,&1<\text{x}\leq4\end{cases}$
$\therefore\ \text{I}=\int\limits^4_0|\text{x}-1|\text{dx}$
$\Rightarrow\text{I}=\int\limits^{1}_0-(\text{x}-1)\text{dx}+\text{I}=\int\limits^{4}_1(\text{x}-1)\text{dx}$
$\Rightarrow\text{I}=\Big[-\frac{\text{x}^2}{2}+\text{x}\big]^1_0+\Big[\frac{\text{x}^2}{2}-\text{x}\big]^4_1$
$\Rightarrow\text{I}=\frac{-1}{2}+1-0+8-4-\frac{1}{2}+1$
$\Rightarrow\text{I}=5$

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