Explanation

(a)
Let $I = \int_{0}^{8}{|x - 5|dx}$
$= \int_{0}^{5}{- (x - 5)dx + \int_{5}^{8}{(x - 5)dx}}$
$= \left\lbrack - \frac{x^{2}}{2} + 5x \right\rbrack_{0}^{5} + \left\lbrack \frac{x^{2}}{2} - 5x \right\rbrack_{5}^{8}$
$= \left( \frac{25}{2} \right) + \left( - \frac{16}{2} + \frac{25}{2} \right) = 17$