Question

Evaluate the line integral \oint_{\mathcal{C}} 4(y-cos (y)) {d} x+4 sin (y) x {~d} y, \] where \mathcal{C} \) is the circle (y-3)^{2}+(x-2)^{2}=9 \) oriented clockwise.

Answer 1

36π

Explanation:

`\mathcal{C}:(y-3)^(2)+(x-2)^(2)=9 \)`

`\displaystyle \oint P\,dx+Q\,dy =\int_D\biggr((\partial Q)/(\partial x)-(\partial P)/(\partial y)\biggr)\,dA`

`P=4(y-\cos(y))=4y-4\cos(y)\\(\partial P)/(\partial y)=4+4\sin(y)`

`Q=4\sin(y)x\\(\partial Q)/(\partial x)=4\sin(y)`

Clockwise Orientation

`\displaystyle -\int_D(4\sin(y)-(4+4\sin(y))\,dA\\\\=-\int_D-4\,dA\\\\=\int_D4\,dA\\\\=\int_0^(2\pi)\int_0^34r\,dr\,d\theta\\\\=\int_0^(2\pi)18\,d\theta\\\\=18(2\pi)\\\\=36\pi`