Use Green's Theorem to evaluate oint_C(x^2+y)dx-(3x+y^3)dy Where c is the ellipse x^2+4y^2=4

Use Green's Theorem to evaluate oint_C(x^2+y)dx-(3x+y^3)dy Where c is the ellipse x^2+4y^2=4

Question
Use Green's Theorem to evaluate
\(\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}+{y}^{{3}}\right)}{\left.{d}{y}\right.}\)
Where c is the ellipse \(\displaystyle{x}^{{2}}+{4}{y}^{{2}}={4}\)

Answers (1)

2021-02-22
Step 1
It is required to calculate the value of the integral using green’s theorem:
\(\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}-{y}^{{3}}\right)}{\left.{d}{y}\right.}{w}{h}{e}{r}{e}{c}{i}{s}{e}{l}{l}{i}{p}{s}{e}{x}^{{2}}+{4}{y}^{{2}}={4}\)
Step 2
\(\displaystyle\oint_{{C}}{P}{\left.{d}{x}\right.}+{Q}{\left.{d}{y}\right.}=\int\int_{{R}}{\left(\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}\right)}{d}{A}\)
Step 3
So,
\(\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}+{y}^{{3}}\right)}{\left.{d}{y}\right.}=\int\int_{{R}}\frac{\partial}{{\partial{x}}}{\left(-{\left({3}{x}+{y}^{{3}}\right)}\right)}-\frac{\partial}{{\partial{y}}}{\left({x}^{{2}}+{y}\right)}\)
\(\displaystyle=\int\int_{{R}}{\left(-{3}-{1}\right)}{d}{A}\)
\(\displaystyle=\int\int_{{R}}{\left(-{4}\right)}{d}{A}\)
Step 4
\(\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}+{y}^{{3}}\right)}{\left.{d}{y}\right.}=\int\int_{{R}}{\left(-{4}\right)}{d}{A}={\int_{{-{{1}}}}^{{1}}}{\int_{{-{{2}}}}^{{2}}}{\left(-{4}\right)}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)
\(\displaystyle={\left(-{4}\right)}{\int_{{-{{1}}}}^{{1}}}{{\left[{x}\right]}_{{-{{2}}}}^{{2}}}{\left.{d}{y}\right.}\)
\(\displaystyle={\left(-{4}\right)}{\int_{{-{{1}}}}^{{1}}}{\left({2}-{\left(-{2}\right)}\right)}{\left.{d}{y}\right.}\)
\(\displaystyle={\left(-{4}\right)}{\int_{{-{{1}}}}^{{1}}}{4}{\left.{d}{y}\right.}\)
\(\displaystyle={\left(-{16}\right)}{{\left({y}\right)}_{{-{{1}}}}^{{1}}}\)
=-16(1-(-1))
=-16(2)
\(\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}^{+}{y}^{{3}}\right)}{\left.{d}{y}\right.}=-{32}\)
0

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