Example 1. The velocity of a body changes according to the law $v=\frac{1}{2} t^{2}$ (m/s). What distance will the body travel in 12 s? What is the speed of motion?
Solution. The path is equal to the integral of the speed:

$$
S=\int_{0}^{12} \frac{1}{2} t^{2} d t=\left.\frac{1}{6} t^{3}\right|_{0} ^{12}=288(M)
$$

The average speed is $v_{\mathrm{cp}}=\frac{S}{t}=\frac{288}{12}=24(\mathrm{m} / \mathrm{c})$.

Answer. $S=288 \mathrm{~m} ; v_{\text {avg }}=24 \mathrm{~m} / \mathrm{c}$.