⃗(x, y)=(3x−⃗ +⃗ f→(x, y)=(3x−4y)i→+2xj→ and 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 .

By Green's theorem, the integral of along is

which is 6 times the area of , the region with 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 and radius , the area of the sector is proportional to the circle's overall area according to

so that the value of the integral is

example 2 – simplify: step 1: completely factor both the numerators and denominators of all fractions. step 2: change the division sign to a multiplication sign and flip (or reciprocate) the fraction after the division sign; essential you need to multiply by the reciprocal. step 3 : cancel or reduce the fractions.

step-by-step explanation: