Question

Let ????C be the positively oriented circle x2+y2=1x2+y2=1. Use Green's Theorem to evaluate the line integral ∫????7y????x+6x????y∫C7ydx+6xdy. equation editorEquation Editor

Answer 1

By Green's theorem,

$\displaystyle\int_C7y\,\mathrm dx+6x\,\mathrm dy$

$=\displaystyle\iint_{x^2+y^2\le1}\frac{\partial(6x)}{\partial x}-\frac{\partial(7y)}{\partial y}\,\mathrm dx\,\mathrm dy$

$=-\displaystyle\iint_{x^2+y^2\le1}\mathrm dx\,\mathrm dy$

which is -1 times the area of the disk $x^2+y^2\le1$. Recall the area of a circle with radius $r$ is $\pi r^2$, so the integral has a value of $\boxed{-\pi}$.