Question

Use​ Green's Theorem to evaluate the line integral. Assume the curve is oriented counterclockwise. Contour integral Subscript Upper C Superscript Baseline (7 x plus cosine StartFraction 1 Over y EndFraction )dy minus (3 y squared plus ln 3 x )dx ​, where C is the boundary of the square with vertices (1 comma 0 )comma (4 comma 0 )comma (4 comma 3 )comma and (1 comma 3 ). Contour integral Subscript Upper C Superscript Baseline (7 x plus cosine StartFraction 1 Over y EndFraction )dy minus (3 y squared plus ln 3 x )dx ​= nothing ​(Type an exact​ answer.)

Answer 1

By Green's theorem, we have

$\displaystyle\int_C\left(7x+\cos\frac1y\right)\,\mathrm dy-(3y^2+\ln(3x))\,\mathrm dx$

$=\displaystyle\iint_{[1,4]\times[0,3]}\frac{\partial\left(7x+\cos\frac1y\right)}{\partial x}-\frac{\partial(-(3y^2+\ln(3x))}{\partial y}\,\mathrm dx\,\mathrm dy$

$=\displaystyle\int_0^3\int_1^47+6y\,\mathrm dx\,\mathrm dy=\boxed{144}$