7. (15 points) Use Green's theorem to evaluate the integral
\[
\oint_{C}\left(x^{2}-y^{2}\right) d x+x y d y
\]
where $C$ is the boundary of the region bounded by $x=y^{2}$ and $x=1$, and orientation of $C$ is counterclockwise.

Step 1 ：First, we need to identify the vector field \(\mathbf{F} = (x^2 - y^2, xy)\).

Step 2 ：Next, we calculate the partial derivatives of the components of \(\mathbf{F}\) to apply Green's theorem. The partial derivative of \(x^2 - y^2\) with respect to \(y\) is \(-2y\), and the partial derivative of \(xy\) with respect to \(x\) is \(y\).

Step 3 ：Green's theorem states that the line integral around a simple closed curve \(C\) of a vector field is equal to the double integral over the region \(D\) enclosed by \(C\) of the divergence of the vector field. In this case, the divergence of \(\mathbf{F}\) is \(\frac{\partial}{\partial x}(x^2 - y^2) + \frac{\partial}{\partial y}(xy) = -2y + y = -y\).

Step 4 ：The region \(D\) is bounded by the curves \(x = y^2\) and \(x = 1\). In terms of \(y\), this region is bounded by \(y = -1\) and \(y = 1\).

Step 5 ：We can now set up the double integral over \(D\) of the divergence of \(\mathbf{F}\): \(\int_{-1}^{1} \int_{y^2}^{1} -y \, dx \, dy\).

Step 6 ：First, we integrate with respect to \(x\): \(\int_{-1}^{1} [-y(x - y^2)]_{y^2}^{1} \, dy = \int_{-1}^{1} -y + y^3 \, dy\).

Step 7 ：Next, we integrate with respect to \(y\): \([-\frac{1}{2}y^2 + \frac{1}{4}y^4]_{-1}^{1} = -\frac{1}{2} + \frac{1}{4} - \left(-\frac{1}{2} - \frac{1}{4}\right) = 0\).

Step 8 ：Therefore, by Green's theorem, the line integral of \(\mathbf{F}\) around \(C\) is \(\boxed{0}\).