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Use the Green's Theorem to calculate the work done by the field

[ F (x, y) = -3y^5 i + 5y^2x^3 j ] to move a particle along the circumference
[C: x^2 + y^2 = 9] starting from the point (2;0) and arriving at the point (-2,0).

User FuePi
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1 Answer

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Let
R be the region bounded by
C. By Green's theorem,


\displaystyle\int_C\mathbf F\cdot\mathrm d\,\mathbf r=\iint_R\left((\partial Q)/(\partial x)-(\partial P)/(\partial y)\right)\,\mathrm dA

where
\mathbf F(x,y)=P(x,y)\,\mathbf i+Q(x,y)\,\mathbf j. Expressing the area in polar coordinates, you have


(\partial Q)/(\partial x)=15x^2y^2=15r^4\sin^2\theta\cos^2\theta

(\partial P)/(\partial y)=-15y^4=-15r^4\sin^4\theta


\displaystyle\int_(\theta=0)^(\theta=\pi)\int_(r=0)^(r=3)\left(15r^3\sin^2\theta\cos^2\theta+15r^4\sin^4\theta\right)r\,\mathrm dr\,\mathrm d\theta

=\displaystyle15\left(\int_0^\pi\sin^2\theta\,\mathrm d\theta\right)\left(\int_0^3r^5\,\mathrm dr\right)

=\frac{3645\pi}4
User Taras Kalapun
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