Find the area bounded by the parabola $y = x^2 - 4$, the X axis and the lines $x = -1$ and $x = 2$.
Given: $y = x^2 - 4$ $\;\;\; \cdots \; (1)$
Required area (shaded portion in the figure) $= \left|I\right|$ where
$\begin{aligned}
I & = \int \limits_{-1}^{2} y \; dx \\\\
& = \int \limits_{-1}^{2} \left(x^2 - 4\right) \; dx \hspace{3em} \left[\text{From equation }(1) \right] \\\\
& = \left[\dfrac{x^3}{3} - 4 x\right]_{-1}^{2} \\\\
& = \dfrac{2^3}{3} - 4 \times 2 - \dfrac{\left(-1\right)^3}{3} + 4 \times \left(-1\right) \\\\
& = - 9
\end{aligned}$
$\therefore$ Required area $= \left|I\right| = 9$ square units