Question

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

∫???? (3y + 7e√x) dx + (8x + 9 cosy²) dy
where C is the boundary of the region enclosed by the parabolas y = x² and x = y².

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)(\partial(8x+9\cos(y^2)))/(\partial x)-(\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}`