Step 1

\(\displaystyle\oint_{{C}}{\left({x}-{y}\right)}{\left.{d}{x}\right.}+{\left({x}+{y}\right)}{\left.{d}{y}\right.}\)

C is along a curve

\(\displaystyle{x}^{{2}}+{y}^{{2}}={9}\)

\(\displaystyle{x}={3}{\cos{{0}}}\)

\(\displaystyle{y}={3}{\sin{{0}}}\)

0 varies from 0 to 2 \(\displaystyle\pi\)

\(\displaystyle\oint-{C}{\left({3}{\cos{{0}}}-{3}{\sin{{0}}}\right)}\cdot-{3}{\sin{{0}}}{d}{0}+{\left({3}{\cos{{0}}}+{3}{\sin{{0}}}\right)}{3}{\cos{{0}}}{d}{0}\)

\(\displaystyle\oint_{{C}}{9}{\left(-{\cos{{o}}}{\sin{{o}}}+{{\sin}^{{2}}{0}}+{{\cos}^{{2}}{0}}+{\cos{{0}}}{\sin{{0}}}\right)}{d}{0}\)

\(\displaystyle{9}\oint_{{C}}{d}{0}\)

\(\displaystyle{9}{\int_{{0}}^{{{2}\pi}}}{d}{0}\)

\(\displaystyle{9}\cdot{2}\pi\)

\(\displaystyle={18}\pi\)

Step 2

Using green's theorem

\(\displaystyle\oint_{{C}}{\left({x}-{y}\right)}{\left.{d}{x}\right.}+{\left({x}+{y}\right)}{\left.{d}{y}\right.}=\)

\(\displaystyle\int\int_{{C}}{2}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)

\(\displaystyle{x}={r}{\cos{{0}}}\)

\(\displaystyle{y}={r}{\sin{{0}}}\)

r varies from 0 to 3

0 varies from 0 to 2 \(\displaystyle\pi\)

\(\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{\int_{{0}}^{{3}}}{r}{d}{r}{d}{0}\)

\(\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{{\left(\frac{{r}^{{2}}}{{2}}\right)}_{{0}}^{{3}}}{d}{0}\)

\(\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}\frac{{9}}{{2}}{d}{0}\)

\(\displaystyle{9}{{\left({0}\right)}_{{0}}^{{{2}\pi}}}\)

\(\displaystyle{18}\pi\)

\(\displaystyle\oint_{{C}}{\left({x}-{y}\right)}{\left.{d}{x}\right.}+{\left({x}+{y}\right)}{\left.{d}{y}\right.}\)

C is along a curve

\(\displaystyle{x}^{{2}}+{y}^{{2}}={9}\)

\(\displaystyle{x}={3}{\cos{{0}}}\)

\(\displaystyle{y}={3}{\sin{{0}}}\)

0 varies from 0 to 2 \(\displaystyle\pi\)

\(\displaystyle\oint-{C}{\left({3}{\cos{{0}}}-{3}{\sin{{0}}}\right)}\cdot-{3}{\sin{{0}}}{d}{0}+{\left({3}{\cos{{0}}}+{3}{\sin{{0}}}\right)}{3}{\cos{{0}}}{d}{0}\)

\(\displaystyle\oint_{{C}}{9}{\left(-{\cos{{o}}}{\sin{{o}}}+{{\sin}^{{2}}{0}}+{{\cos}^{{2}}{0}}+{\cos{{0}}}{\sin{{0}}}\right)}{d}{0}\)

\(\displaystyle{9}\oint_{{C}}{d}{0}\)

\(\displaystyle{9}{\int_{{0}}^{{{2}\pi}}}{d}{0}\)

\(\displaystyle{9}\cdot{2}\pi\)

\(\displaystyle={18}\pi\)

Step 2

Using green's theorem

\(\displaystyle\oint_{{C}}{\left({x}-{y}\right)}{\left.{d}{x}\right.}+{\left({x}+{y}\right)}{\left.{d}{y}\right.}=\)

\(\displaystyle\int\int_{{C}}{2}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)

\(\displaystyle{x}={r}{\cos{{0}}}\)

\(\displaystyle{y}={r}{\sin{{0}}}\)

r varies from 0 to 3

0 varies from 0 to 2 \(\displaystyle\pi\)

\(\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{\int_{{0}}^{{3}}}{r}{d}{r}{d}{0}\)

\(\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{{\left(\frac{{r}^{{2}}}{{2}}\right)}_{{0}}^{{3}}}{d}{0}\)

\(\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}\frac{{9}}{{2}}{d}{0}\)

\(\displaystyle{9}{{\left({0}\right)}_{{0}}^{{{2}\pi}}}\)

\(\displaystyle{18}\pi\)