Explanation

(a)
$\int_{- 2}^{3}{\left| 1 - x^{2} \right|dx = \int_{- 2}^{- 1}{\left( x^{2} - 1 \right)dx}}$
$+ \int_{- 1}^{1}{\left( 1 - x^{2} \right)dx + \int_{1}^{3}{\left( x^{2} - 1 \right)dx}}$
$= \left\lbrack \frac{x^{3}}{3} - x \right\rbrack_{- 2}^{- 1} + \left\lbrack x + \frac{x^{3}}{3} \right\rbrack_{- 1}^{1} + \left\lbrack \frac{x^{3}}{3} - x \right\rbrack_{1}^{3}$
$= \frac{4}{3} + \frac{4}{3} + \frac{20}{3} = \frac{28}{3}$