362. Calculate the integral $\int_{1}^{4} x^{2} d x$ using Simpson's formula.
Solution. Divide the interval of integration into 10 equal parts. Then $(b-a) / 3 n=3 / 30=1 / 10=0.1$. Substituting into the integrand function $y=x^{2}$ the values of the argument $x_{0}=1, x_{1}=1.3 ; x_{2}=1.6, \ldots$, $x_{10}=4$, we find the corresponding values of the ordinates: $y_{0}=1 ; y_{1}=1.69$; $y_{2}=2.56 ; \quad y_{3}=3.61 ; \quad y_{4}=4.84 ; \quad y_{5}=6.25 ; \quad y_{6}=7.84 ; \quad y_{7}=9.61 ; \quad y_{8}=11.56 ;$ $y_{9}=13.69 ; y_{10}=16$.

Applying formula (14), we get

$$
\begin{gathered}
\int_{1}^{4} x^{2} d x=0.1((1+16)+2(2.56+4.84+7.84+11.56)+4(1.69+3.61+6.25+ \\
+9.61+13.69))=21
\end{gathered}
$$

Evaluating the integral using the Newton-Leibniz formula gives

$$
\int_{1}^{4} x^{2} d x=\left.\frac{x^{3}}{3}\right|_{1} ^{4}=\frac{64}{3}-\frac{1}{3}=21
$$

Thus, by applying Simpson's formula, in this case, we obtained the exact value of the integral.

Note that although the method of rectangles is the simplest method for approximate calculation of definite integrals, it provides the least accurate results. The choice of the method of approximate integration depends on the integrand function and the required accuracy of the calculation.

## $\S$ 10. Application of the Definite Integral to Solving Physical Problems

Scheme for solving problems on the application of the definite integral Finding the path traveled by a body in rectilinear motion Calculating the work done by a force in rectilinear motion of a body Calculating the work expended on stretching or compressing a spring Determining the force of liquid pressure on a vertically positioned plate