Evaluate using Greenโs Theorem โฎ 3๐ฅ๐ฆ๐๐ฅ + 2๐ฅ๐ฆ๐๐ฆ, where ๐ถ is the rectangle bounded by
๐ฅ = โ2, ๐ฅ = 4, ๐ฆ = 1 and ๐ฆ = 2.
โฎ3xydx+2xydy=โฌ(โโx(2xy)โโโy(3xy))dxdy==โฌ(2yโ3x)dxdy=โซโ24โซ12(2yโ3x)dydx=โซโ24(3โ3x)dx=(3xโ32x2)โฃโ24==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}=\\=0โฎ3xydx+2xydy=โฌ(โxโโ(2xy)โโyโโ(3xy))dxdy==โฌ(2yโ3x)dxdy=โซโ24โโซ12โ(2yโ3x)dydx=โซโ24โ(3โ3x)dx=(3xโ23โx2)โฃโ24โ==0
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