VC5.8

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Let R be the region bounded by C,thus by Green's theorm,

\oint _{C}[(\cos x\sin y-xy)dx+\sin x\cos ydy]\,

=\iint _{R}[{\frac  {\partial }{\partial x}}(\sin x\cos y)-{\frac  {\partial }{\partial y}}(\cos x\sin y-xy)]dxdy\,

=\iint _{R}(\cos x\cos y-\cos x\cos y+x)dxdy\,

=\iint _{R}xdxdy=\iint _{R}(r\cos \theta )(rd\theta dr)\, [on changing to polar coordinates]

=\int _{{0}}^{{2\pi }}\int _{{r=0}}^{{1}}r^{2}\cos \theta d\theta dr\,, since for the region R,r varies from 0 to 1 and theta varies from 0 to 2pi.

=\int _{{0}}^{{2\pi }}\cos \theta [{\frac  {r^{3}}{3}}]_{0}^{1}d\theta \,

={\frac  {1}{3}}[\sin \theta ]_{{0}}^{{2\pi }}=0\,

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