VC5.11

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By Green's theorm,\oint _{C}[(xy+y^{2})dx+x^{2}dy]=\oint _{C}[{\frac  {\partial }{\partial x}}(x^{2})-{\frac  {\partial }{\partial y}}(xy+y^{2})]dxdy\, --(1)

We evaluate LHS of the above,on y=x^{2},dy=2xdx\, and x varies from 0 to 1,and on y=x,dy=dx and x varies from 1 to 0.

LHS=\int _{0}^{1}[(x\cdot x^{2}+x^{4})dx+x^{2}(2xdx)]+\int _{1}^{0}[(x\cdot x+x^{2})dx+x^{2}dx]\,

=\int _{0}^{1}(3x^{3}+x^{4})dx+3\int _{1}^{0}x^{2}dx={\frac  {19}{20}}-1={\frac  {-1}{20}}\,

RHS=\int _{{x=0}}^{{1}}\int _{{y=x^{2}}}^{{y=x}}(x-2y)dxdy=\int _{0}^{1}[xy-y^{2}]_{{x^{2}}}^{{x}}dx\,

=\int _{0}^{1}(x^{2}-x^{2}-x^{3}+x^{4})dx=\int _{0}^{1}(x^{4}-x^{3})dx={\frac  {-1}{20}}\, which is equal to LHS.

Hence the theorm is verified.

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