Question

using Greens theorem in the plane evaluate ∮(3x+4y)dx+(2x-3y)dy where C is the circle x²+y²=4 traversed in the counterclockwise sense

Answer 1

$$P(x, y)=3x+4y, \dfrac{\partial P}{\partial y}=4$$

$$Q(x, y)=2x-y, \dfrac{\partial Q}{\partial x}=2$$

Applying Green’s Theorem, you then have

$$∮_C(3x+4y)dx+(2x-3y)dy$$

$$=\int \int _D(2-4)dxdy$$

Transforming to polar coordinates, we obtain

$$∮_C(3x+4y)dx+(2x-3y)dy$$

$$=\int \int _D(2-4)dxdy$$

$$=\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{2}(-2)rdrd\theta$$

$$=-2\pi(2)^2=-8\pi$$