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.

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.

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 $$\displaystyle\pi$$
$$\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{\int_{{0}}^{{3}}}{r}{d}{r}{d}{0}$$
$$\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{{\left(\frac{{r}^{{2}}}{{2}}\right)}_{{0}}^{{3}}}{d}{0}$$
$$\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}\frac{{9}}{{2}}{d}{0}$$
$$\displaystyle{9}{{\left({0}\right)}_{{0}}^{{{2}\pi}}}$$
$$\displaystyle{18}\pi$$

Relevant Questions

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