Question #314526

Evaluate using Greenโ€™s Theorem โˆฎ 3๐‘ฅ๐‘ฆ๐‘‘๐‘ฅ + 2๐‘ฅ๐‘ฆ๐‘‘๐‘ฆ, where ๐ถ is the rectangle bounded by


๐‘ฅ = โˆ’2, ๐‘ฅ = 4, ๐‘ฆ = 1 and ๐‘ฆ = 2.


1
Expert's answer
2022-03-21T11:21:47-0400

โˆฎ3xydx+2xydy=โˆฌ(โˆ‚โˆ‚x(2xy)โˆ’โˆ‚โˆ‚y(3xy))dxdy==โˆฌ(2yโˆ’3x)dxdy=โˆซโˆ’24โˆซ12(2yโˆ’3x)dydx=โˆซโˆ’24(3โˆ’3x)dx=(3xโˆ’32x2)โˆฃโˆ’24==0\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


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