Example 5. Find the area of the figure bounded by the lines:

$$
y=x^{2}, y^{2}=x
$$
Solution. Solving the system of equations $y=x^{2}, y^{2}=x$ simultaneously, we find the abscissas of the points of intersection of these curves: $x_{1}=0, x_{2}=1$. Therefore, the limits of integration will be: $a=0, b=1$ (Fig. 5.9). The required area is calculated using formula (5.15). First, note that $y_{1}=\sqrt{x}, y_{2}=x^{2}$ (obtained from the equations of the lines).

Thus,

$$
\begin{aligned}
S=\int_{0}^{1}\left(y_{1}-y_{2}\right) d x & =\int_{0}^{1}\left(\sqrt{x}-x^{2}\right) d x=\int_{0}^{1} \sqrt{x} d x-\int_{0}^{1} x^{2} d x= \\
& =\left.\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right|_{0} ^{1}-\left.\frac{x^{3}}{3}\right|_{0} ^{1}=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-211.jpg?height=449&width=610&top_left_y=569&top_left_x=55)

Fig. 5.10