Use green’s theorem to evaluate the integral i = z c 2y 3 dx − 2x 3 dy where c is the circle of radius 2 centered at the origin, oriented counterclockwise

-48π

Step-by-step explanation:

Recall that Green's theorem states the following:

Let P, Q be -functions on a region A, which is the interior of a closed piecewise -path C, parametrized counterclockwise. Then  

We have

Therefore

where A is the interior of the circle of radius 2 centered at the origin.

If we change to polar coordinates

x = rcos(t)

y = rsin(t)

Then A can be described with the inequalities

0 < r < 2

0 < t < 2π  

and

step-by-step explanation:

could you provide figure.