Explanation

(b)
Given, $\int_{0}^{a}{x\ dx \leq a + 4}$
$\Rightarrow \ \left\lbrack \frac{x^{2}}{2} \right\rbrack_{0}^{a} \leq a + 4\ \Rightarrow \frac{a^{2}}{2} \leq a + 4$
$\Rightarrow \ a^{2} \leq 2a + 8\ \Rightarrow a^{2} - 2a - 8 \leq 0$
$\Rightarrow \ (a - 4)(a + 2) \leq 0 \Rightarrow \ - 2 \leq a \leq 4$