Question

The integral $\int\limits_{0}^{2} \| x-1|-x| d x$ is equal to ______.

Answer 1

$\int\limits_{0}^{2}|x-1|-x \mid d x$
Let $f(x) \| x-1|-x|=\left\{\begin{array}{ll}1, & x \geq 1 \\ |1-2 x|, & x \leq 1\end{array}\right.$

$A=\frac{1}{2}+1=\frac{3}{2}$
Or
$\int\limits_{0}^{1 / 2}(1-2 x) d x+\int\limits_{1 / 2}^{1}(2 x-1)+\int\limits_{0}^{2} 1 d x$
$=\left[x-x^{2}\right]_{0}^{\frac{1}{2}}+\left[x^{2}-x\right]_{1 / 2}^{1}+[x]_{1}^{2}$
$=3 / 2$