Explanation

(b)
$\int_{- 1}^{1}{\max{\left( \left\{ x \right\},\ \left\{ - x \right\} \right)dx}}$
$= \int_{- 1}^{- 0.5}{\left\{ - x \right\} dx + \int_{- 0.5}^{0}{\left\{ x \right\} dx +}\int_{0}^{0.5}{\left\{ - x \right\} dx +}\int_{0.5}^{1}{\left\{ x \right\} dx}}$
$= \int_{- 1}^{- 0.5}{\left\{ - x \right\} dx +}\int_{- 0.5}^{0}{\left\{ x - \lbrack x\rbrack \right\} dx +}\int_{0}^{0.5}{\left( - x - \lbrack - x\rbrack \right)dx + \int_{0.5}^{1}{\left( x - \lbrack x\rbrack \right)dx}}$
$= \left\lbrack - \frac{x^{2}}{2} \right\rbrack_{- 1}^{- 0.5} + \left\lbrack \frac{x^{2}}{2} + x \right\rbrack_{- 0.5}^{0} + \left\lbrack x - \frac{x^{2}}{2} \right\rbrack_{0}^{0.5} + \left\lbrack \frac{x^{2}}{2} \right\rbrack_{0.5}^{1}$
$= - \frac{0.25}{2} + \frac{1}{2} + \left( - \frac{0.25}{2} + 0.5 \right) + 0.5 - \frac{0.25}{2} + \frac{1}{2} - \frac{0.25}{2}$
$= 2 - \frac{1}{2} = \frac{3}{2}$