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
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Tuthornt

Answered 2021-01-16
Author has **107** answers

Step 1

${\oint}_{C}(x-y)dx+(x+y)dy$

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$\pi$

$\oint -C(3\mathrm{cos}0-3\mathrm{sin}0)\cdot -3\mathrm{sin}0d0+(3\mathrm{cos}0+3\mathrm{sin}0)3\mathrm{cos}0d0$

${\oint}_{C}9(-\mathrm{cos}o\mathrm{sin}o+{\mathrm{sin}}^{2}0+{\mathrm{cos}}^{2}0+\mathrm{cos}0\mathrm{sin}0)d0$

$9{\oint}_{C}d0$

$9{\int}_{0}^{2\pi}d0$

$9\cdot 2\pi$

$=18\pi$

Step 2

Using greens

C is along a curve

0 varies from 0 to 2

Step 2

Using greens

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