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.

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.

Expert Answers (1)

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 \(\displaystyle\pi\)
\(\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{\int_{{0}}^{{3}}}{r}{d}{r}{d}{0}\)
\(\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}{{\left(\frac{{r}^{{2}}}{{2}}\right)}_{{0}}^{{3}}}{d}{0}\)
\(\displaystyle{2}{\int_{{0}}^{{{2}\pi}}}\frac{{9}}{{2}}{d}{0}\)
\(\displaystyle{9}{{\left({0}\right)}_{{0}}^{{{2}\pi}}}\)
\(\displaystyle{18}\pi\)
7
 
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