Example 2.47. Calculate the area of the figure bounded by the parabolas $y=x^{2}$ and $y=8-x^{2}$ (Fig. 2.2).
Solution. To find the points of intersection of the parabolas, we will solve the system of equations $y=x^{2}$ and $y=8-x^{2}$

$$
\Rightarrow x_{1}=-2, y_{1}=4, x_{2}=2, y_{2}=4
$$

We will use formula (2.20) due to the evenness of the integrand:

$$
\begin{gathered}
S=\int_{-2}^{2}\left[\left(8-x^{2}\right)-x^{2}\right] d x=\int_{0}^{2}\left[\left(8-x^{2}\right)-x^{2}\right] d x= \\
=\left.\left[16 x-\frac{4}{3} x^{3}\right]\right|_{0} ^{2}=\frac{64}{3}=31 \frac{1}{3}
\end{gathered}
$$