Question

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.

Answer 1

By Green's theorem, the line integral is equivalent to the area integral

$\displaystyle\int_C(3y+7e^{\sqrt x})\,\mathrm dx+(8x+9\cos(y^2))\,\mathrm dy=\int_0^1\int_{x^2}^{\sqrt x}\frac{\partial(8x+9\cos(y^2))}{\partial x}-\frac{\partial(3y+7e^{\sqrt x})}{\partial y}\,\mathrm dy\,\mathrm dx$

$=\displaystyle5\int_0^1\int_{x^2}^{\sqrt x}\mathrm dy\,\mathrm dx=\boxed{\frac53}$