Example 8. Check the conditions of Green's theorem for the line integral $\int_{L} 2 x y d x + x^{2} d y$ and compute this integral along the parabola $y=\frac{x^{2}}{4}$ from the origin to the point $A(2,1)$.
Solution. We have $P(x, y)=2 x y, Q(x, y)=x^{2}$. These functions are defined, continuous, and differentiable at any point $(x, y)$ in the plane. We have $\frac{\partial P}{\partial y}=2 x, \frac{\partial Q}{\partial x}=2 x$. The conditions of Green's theorem are satisfied. Therefore, the given integral is independent of the path of integration. Moreover, the expression $2 x y d x+x^{2} d y$ represents the total differential of some function $U(x, y)$. It is not difficult to guess that $U(x, y)=x^{2} y$. Therefore,

$$
\int_{O A} 2 x y d x+x^{2} d y=\left.x^{2} y\right|_{(0,0)} ^{(2, \mathrm{I})}=4
$$

Direct calculation gives $\left(y=\frac{x^{2}}{4}, d y=\frac{x}{2} d x\right)$ :

$$
\int_{O A} 2 x y d x+x^{2} d y=\int_{0}^{2}\left(\frac{x^{3}}{2}+\frac{x^{3}}{2}\right) d x=\left.\frac{x^{4}}{4}\right|_{0} ^{2}=4
$$

The answers match.

Answer. 4.