Answer to Question #314526 in Calculus for Jyo

Question

Evaluate using Green’s Theorem ∮ 3𝑥𝑦𝑑𝑥 + 2𝑥𝑦𝑑𝑦, where 𝐶 is the rectangle bounded by

𝑥 = −2, 𝑥 = 4, 𝑦 = 1 and 𝑦 = 2.

Answer 1

$\oint{3xydx+2xydy}=\iint{\left( \frac{\partial}{\partial x}\left( 2xy \right) -\frac{\partial}{\partial y}\left( 3xy \right) \right) dxdy}=\\=\iint{\left( 2y-3x \right) dxdy}=\int_{-2}^4{\int_1^2{\left( 2y-3x \right) dydx}}=\int_{-2}^4{\left( 3-3x \right) dx}=\left( 3x-\frac{3}{2}x^2 \right) |_{-2}^{4}=\\=0$