Question

????⃗ (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.

Answer 1

By Green's theorem, the integral of $\vec F$ along $C$ is

$\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_D\left(\frac{\partial(2x)}{\partial x}-\frac{\partial(3x-4y)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=6\iint_D\mathrm dx\,\mathrm dy$

which is 6 times the area of $D$, the region with $C$ 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 $\theta$ and radius $r$, the area $A$ of the sector is proportional to the circle's overall area according to

$\dfrac A{\frac\pi3\,\rm rad}=\dfrac{16\pi}{2\pi\,\rm rad}\implies A=\dfrac{8\pi}3$

so that the value of the integral is

$\dfrac{6\times8\pi}3=\boxed{16\pi}$