Question

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C 3y + 5e x dx + 8x + 9 cos(y2) dy C is the boundary of the region enclosed by the parabolas y = x2 and x = y2

Answer 1

By Green's theorem,

$\displaystyle\int_C(3y+5e^x)\,\mathrm dx+(8x+9\cos(y^2))\,\mathrm dy$

$\displaystyle=\iint_D\left(\frac{\partial(8x+9\cos(y^2))}{\partial x}-\frac{\partial(3y+5e^x)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=5\iint_D\mathrm dx\,\mathrm dy$

where $D$ is the region enclosed by $C$. Equivalently, the line integral is equal to 5 times the area of $D$, which is

$\displaystyle5\iint_D\mathrm dx\,\mathrm dy=5\int_0^1\int_{x^2}^{\sqrt x}\mathrm dy\,\mathrm dx=5\int_0^1(\sqrt x-x^2)\,\mathrm dx=\boxed{\frac53}$