Textbook Problem

Verifying Green’s TheoremIn Exercises 9 and 10, verify Green’s Theorem by using a computer algebra system to evaluate both integrals ${\displaystyle {\int }_{c}x{e}^{y}dx+e^{x}dy}={\displaystyle {\int }_R{\displaystyle \int \left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)}dA}$ for the given path.

C: boundary of the region lying between the graphs of $y=x$ and $y=x^3$ in the first quadrant