 # 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. babeeb0oL 2021-01-15 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.
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Step 1
${\oint }_{C}\left(x-y\right)dx+\left(x+y\right)dy$
C is along a curve
${x}^{2}+{y}^{2}=9$
$x=3\mathrm{cos}0$
$y=3\mathrm{sin}0$
0 varies from 0 to 2 $\pi$
$\oint -C\left(3\mathrm{cos}0-3\mathrm{sin}0\right)\cdot -3\mathrm{sin}0d0+\left(3\mathrm{cos}0+3\mathrm{sin}0\right)3\mathrm{cos}0d0$
${\oint }_{C}9\left(-\mathrm{cos}o\mathrm{sin}o+{\mathrm{sin}}^{2}0+{\mathrm{cos}}^{2}0+\mathrm{cos}0\mathrm{sin}0\right)d0$
$9{\oint }_{C}d0$
$9{\int }_{0}^{2\pi }d0$
$9\cdot 2\pi$
$=18\pi$
Step 2
Using greens