Question # Evaluate the line integral by the two following methods. y) dx + (x+y)dy C os counerclockwise around the circle with center the origin and radius 3(a) directly (b) using Green's Theorem.

Green's, Stokes', and the divergence theorem
ANSWERED Evaluate the line integral by the two following methods. y) dx + (x+y)dy C os counerclockwise around the circle with center the origin and radius 3(a) directly (b) using Green's Theorem. 2021-01-16
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 $$\displaystyle\pi$$
$$\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{\int_{{0}}^{{3}}}{r}{d}{r}{d}{0}$$
$$\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{{\left(\frac{{r}^{{2}}}{{2}}\right)}_{{0}}^{{3}}}{d}{0}$$
$$\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}\frac{{9}}{{2}}{d}{0}$$
$$\displaystyle{9}{{\left({0}\right)}_{{0}}^{{{2}\pi}}}$$
$$\displaystyle{18}\pi$$