Question

Apply Greens Theorem to evaluate the integral. D19y + x)dx + (y + 3x)dy C: The circle (x - 7)2 + (y-7)2 = 6 $19y + xApply​ Green's Theorem to evaluate the integral. ModifyingBelow Contour integral With Upper C left parenthesis 9 y plus x right parenthesis dx plus left parenthesis y plus 3 x right parenthesis dy ∮ C (9y+x)dx+(y+3x)dy     ​C: The circle left parenthesis x minus 7 right parenthesis squared plus left parenthesis y minus 7 right parenthesis squared equals 6 (x−7)2+(y−7)2=6

Answer 1

By Green's theorem,

$\displaystyle\int_C(9y+x)\,\mathrm dx+(y+3x)\,\mathrm dy=\iint_D\frac{\partial(y+3x)}{\partial x}-\frac{\partial(9y+x)}{\partial y}\,\mathrm dy\,\mathrm dx=-6\iint_D\mathrm dy\,\mathrm dx$

where $D$ is the disk with $C$ as its boundary. The integral is simply -6 times the area of the disk $D$, which has radius √6, and hence area 6π, so the value of the integral is -36π.