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Use Green's Theorem to evaluate the line integral along the given positively oriented curve.


Use Green's Theorem to evaluate the line integral

C

5y3 dx ? 5x3 dy

Use Green's Theorem to evaluate the line integral

C is the circle x2 + y2 = 4

Verified Answer :

LammettHash

By Green's theorem,

\(\displaystyle\oint_C(5y^3\,\mathrm dx\pm5x^3\,\mathrm dy)=\iint_D\left(\pm\frac\partial{\partial x}[5x^3]-\frac\partial{\partial y}[5y^3]\right)\,\mathrm dx\,\mathrm dy\)

where \(D\) is the disk of radius 2 centered at the origin.

Differentiating and converting to polar coordinates, you have

\(\displaystyle\iint_D\left(\pm15x^2-15y^2\right)\,\mathrm dx\,\mathrm dy=15\int_0^{2\pi}\int_0^2\left(\pm r^2\cos^2\theta-r^2\sin^2\theta\right)r\,\mathrm dr\,\mathrm d\theta=15\int_0^{2\pi}\int_0^2\left(\pm \cos^2\theta-\sin^2\theta\right)r^3\,\mathrm dr\,\mathrm d\theta\)

which, depending on the sign concealed by the question mark, could be reduced to either

\(\displaystyle15\int_0^{2\pi}\int_0^2r^3\cos2\theta\,\mathrm dr\,\mathrm d\theta=0\)

if a plus, or

\(\displaystyle-15\int_0^{2\pi}\int_0^2r^3\,\mathrm dr\,\mathrm d\theta=-8\pi\)

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