Answer:
-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
In this exercise it will be necessary to use Green's theorem to calculate the perimeter that will be given by
[tex]F= -9\pi[/tex]
Knowing that Green's theorem should be used to solve this exercise then:
[tex]\int\limits_c {4y dx+ 2xy dy} = \int\limits \int\limits {(d/dx)(2xy)-(d/dy)(4y)} \, dA \\= \int\limits \int\limits { 2y-4}\ dA[/tex]
In this way, you must convert to polar coordinates, we will have:
[tex]\int\limits^3_0 \int\limits^{2\pi}_0 {(2rsin(\theta)-4)(r)} \, d\theta dr \\=\int\limits^3_0 {(-2rcos(\theta)-4\theta)(-2r\pi)} \, dr \\=-(\pi)(r^2)= -9\pi[/tex]
We can say that the perimeter of the circle is given by:
[tex]F= -9\pi[/tex]
See more about Green's theorem at brainly.com/question/16793753
