Question

Use Green's Theorem to evaluate F · dr. C (Check the orientation of the curve before applying the theorem.) F(x, y) = e−x + y2, e−y + x2 , C consists of the arc of the curve y = cos(x) from − π 2 , 0 to π 2 , 0 and the line segment from π 2 , 0 to − π 2 , 0

Answer 1

Notice that $C$ is traversed clockwise. Green's theorem applies to curves with a counterclockwise orientation, so we'll have to multiply the area integral by -1.

By Green's theorem, with the vector field $\vec F(x,y)=P(x,y)\,\vec\imath+Q(x,y)\,\vec\jmath$,

$\displaystyle\int_C\vec F\cdot\mathrm d\vec r=-\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,\mathrm dA$

where $D$ is the region with boundary $C$. The partial derivatives are

$\dfrac{\partial(e^{-y}+x^2)}{\partial x}=2x$

$\dfrac{\partial(e^{-x}+y^2)}{\partial y}=2y$

so that the double integral is

$\displaystyle\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,\mathrm dA=\int_{-\pi/2}^{\pi/2}\int_0^{\cos x}2(y-x)\,\mathrm dy\,\mathrm dx=\boxed{\frac\pi2}$