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((\partial(2x))/(\partial x)-(\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

`\frac A{\frac\pi3\,\rm rad}=(16\pi)/(2\pi\,\rm rad)\implies A=\frac{8\pi}3`

so that the value of the integral is

`\frac{6*8\pi}3=\boxed{16\pi}`