????⃗ (x,y)=(3x−4y)????⃗ +2x????⃗ F→(x,y)=(3x−4y)i→+2xj→ and ????C is the counter-clockwise oriented sector of a circle centered at the origin with radius 44 and central angle ????/3π/3. Use Green's theorem to calculate the circulation of ????⃗ F→ around ????C.

By Green's theorem, the integral of [tex]\vec F[/tex] along [tex]C[/tex] is[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_D\left(\frac{\partial(2x)}{\partial x}-\frac{\partial(3x-4y)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=6\iint_D\mathrm dx\,\mathrm dy[/tex]which is 6 times the area of [tex]D[/tex], the region with [tex]C[/tex] as its boundary.We can compute the integral by converting to polar coordinates, or simply recalling the formula for a circular sector from geometry: Given a sector with central angle [tex]\theta[/tex] and radius [tex]r[/tex], the area [tex]A[/tex] of the sector is proportional to the circle's overall area according to[tex]\dfrac A{\frac\pi3\,\rm rad}=\dfrac{16\pi}{2\pi\,\rm rad}\implies A=\dfrac{8\pi}3[/tex]so that the value of the integral is[tex]\dfrac{6\times8\pi}3=\boxed{16\pi}[/tex]