The area bounded by the X-axis and the curve \(y=x(x-2)(x+1)\) is

\(\begin{array}{ll} & \text { For X-axis, } \\ & y=0 \\ \therefore \quad & x(x-2)(x+1)=0 \\ & \Rightarrow x=0 \text { or } x=2 \text { or } x=-1\end{array}\)


\(\begin{aligned} & \text { Required area }=\int_{-1}^0 y \mathrm{~d} x+\left|\int_0^2 y \mathrm{~d} x\right| \\ & =\int_{-1}^0 x(x-2)(x+1) \mathrm{d} x+\left|\int_0^2 x(x-2)(x+1) \mathrm{d} x\right| \\ & =\int_{-1}^0\left(x^3-x^2-2 x\right) \mathrm{d} x+\left|\int_0^2\left(x^3-x^2-2 x\right) \mathrm{d} x\right| \\ & =\left[\frac{x^4}{4}-\frac{x^3}{3}-x^2\right]_{-1}^0+\left|\left[\frac{x^4}{4}-\frac{x^3}{3}-x^2\right]_0^2\right| \\ & =\frac{5}{12}+\left|-\frac{8}{3}\right| \\ & =\frac{37}{12} \text { sq. units }\end{aligned}\)