Question #229931

Using Green's theorem,evaluate âˆ®c(x+y)dx+ xÂ²dy, C is the triangle

with vertices at (0,0). (2,0) and (2, 4), taken in that order.

Expert's answer

The components of the vector field are

"P\n(\nx\n,\ny\n)\n=\nx^2\n,\nQ\n(\nx\n,\ny\n)\n=\nx\n+\ny"

Using the Greenâ€™s formula

"\u222c_\nR\n \n(\\frac{\n\u2202\nQ}{\n\u2202\nx}\n\u2212\n\\frac{\u2202\nP}{\n\u2202\ny}\n)\nd\nx\nd\ny\n=\n\u222e_\nC\n \nP\nd\nx\n+\nQ\nd\ny"

we transform the line integral into the double integral:

"I\n=\n\u222e_\nC\n \nx^2\nd\ny\n+\n(\nx\n+\ny\n)\nd\nx\n=\n\u222c_\nR\n \n(\n\\frac{\u2202\n(\nx\n+\ny\n)}{\n\u2202\nx}\n\u2212\n\\frac{\u2202\n(\nx\ny\n)}{\n\u2202\ny}\n)\nd\nx\nd\ny\n=\n\u222c_\nR\n \n(\n2x\n\u2212\n1\n)\nd\nx\nd\ny"

"\\int _0^2\\int _0^{2x}\\left(2x-1\\right)dydx\\\\\n=\\int _0^22x\\left(2x-1\\right)dx\\\\\n=\\frac{20}{3}"

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