Answer
$$\frac{9}{4}$$
Work Step by Step
Given $$f(x)=x^{3}-2 x+2, \quad[-1,2] $$
The average value is given by
\begin{align*}
M&=\frac{1}{b-a} \int_{a}^{b} f(x) d x\\
&=\frac{1}{2-(-1)} \int_{-1}^{2}\left(x^{3}-2 x+2\right) d x\\
&=\left.\frac{1}{3}\left(\frac{1}{4} x^{4}-x^{2}+2 x\right)\right|_{-1} ^{2}\\
&=\frac{1}{3}\left[(4-4+4)-\left(\frac{1}{4}-1-2\right)\right]\\
&=\frac{9}{4}
\end{align*}
