Example 2.50. Calculate the area of the figure bounded by the parabolas $x=-3 y^{2}$ and $x=1-4 y^{2}$.
Solution. The figure defined by the given equations is shown in Fig. 2.5. It is symmetric with respect to the $O X$ axis, and its area is conveniently calculated using formula (2.21).

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