Suppose F⃗ (x, y)=4yi⃗ +2xyj⃗ . Use Green's Theorem to calculate the circulation of F⃗ around the perimeter of a circle C of radius 3 centered at the origin and oriented counter-clockwise.

In this exercise it will be necessary to use Green's theorem to calculate the perimeter that will be given by

Knowing that Green's theorem should be used to solve this exercise then:

In this way, you must convert to polar coordinates, we will have:

We can say that the perimeter of the circle is given by:

See more about Green's theorem at link

-9π

Step-by-step explanation:

∫c (4y dx + 2xy dy)

= ∫∫ [(∂/∂x)(2xy) - (∂/∂y)(4y)] dA, by Green's Theorem

= ∫∫ (2y - 4) dA

Now convert to polar coordinates:

∫(r = 0 to 3) ∫(θ = 0 to 2π) (2r sin θ - 4) * (r dθ dr) --- first integration

= ∫(r = 0 to 3) (-2r cos θ - 4θ) * r {for θ = 0 to 2π} dr

= ∫(r = 0 to 3) -2πr dr

= -πr² {for r = 0 to 3}

= -π(3²) - -π(0)²

= -9π

25/6 hope it : )

step-by-step explanation: