Example 2.17. Estimate the upper and lower bounds of the integral $\int_{0}^{2} \sqrt{1+x^{3}} d x$.
Solution. Since the function $f(x)$ is monotonically increasing on the interval $[0,2]$, the minimum and maximum values are achieved at the endpoints of the interval. These values are respectively $m=1, M=3$. We obtain the following estimate for the integral in question:

$$
2 \leqslant \int_{0}^{2} \sqrt{1+x^{3}} d x \leqslant 6
$$