Question

let be the positively oriented square with vertices , , , . use green's theorem to evaluate the line integral

Answer 1

If C be a positively oriented square with vertices (0, 0), (3, 0), (3, 3), (0, 3), then the line integral evaluated using the Green's Theorem is (-81).

As per the question statement, C is a positively oriented square with vertices (0, 0), (3, 0), (3, 3), (0, 3).

We are required to calculate the line integral using the Green's Theorem.

Green's Theorem equation is as follows:

$\int\limits_c ({P} \, dx-{Q} \, dy)=\int\limits\int\limits_D {(\frac{dQ}{dx}-\frac{dP}{dy} }) \, dx$

Comparing, $\int\limits_c ({8y^{2}x } \, dx-{6x^{2} y} \, dy)$ with $\int\limits_c ({P} \, dx-{Q} \, dy)$, we get

(P = 8y²x) and (Q = 6x²y).

Then, $[(\frac{dP}{dy})=16xy]$ and $[(\frac{dQ}{dx})=12xy]$.

Here, D = {(x, y): 0≤x≤3, 0≤y≤3}

Then, using Green's Theorem equation,

$\int\limits_c ({8y^{2}x } \, dx-{6x^{2} y} \, dy) = \int\limits^0_3\int\limits^0_3 {(12xy-16xy)} \,dy dx \\=\int\limits^0_3 ({-4xy}) \, dx \\=\int\limits^0_3 ({-2x})(y^{2} ) \, dx \\\\=\int\limits^0_3 ({-18x}) \, dx \\\\=-81$

• Green's theorem: In vector calculus, Green's Theorem relates a line integral around a simple closed curve "C" to a double integral over the plane region "D" which is bounded by "C".

To learn more about Green's Theorem, click on the link below.

brainly.com/question/28187091

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