VC5.3

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By using Green's theorm,\int _{C}[(x^{2}-xy^{3})dx+(y^{2}-2xy)dy]=\iint _{R}[{\frac  {\partial }{\partial x}}(y^{2}-2xy)-{\frac  {\partial }{\partial y}}(x^{2}-xy^{3})]dxdy\, --(1)

Now,RHS of (1) =\int _{{x=0}}^{{2}}\int _{{y=0}}^{{2}}[-2y+3xy^{2}]dxdy\,

=\int _{{x=0}}^{{2}}[-y^{2}+xy^{3}]_{{y=0}}^{{2}}dx=\int _{0}^{2}(-4+8x)dx=[-4x+4x^{2}]_{0}^{2}=-8+16=8\, --(2)

We now evaluate the LHS of (1).

Along the line from O(0,0) to A(2,0),y=0,dy=0 and x varies from 0 to 2.

Along the line from A(2,0) to B(2,2),x=2,dx=0 and y varies from 0 to 2.

Along the line from B(2,2) to C(0,2),y=2,dy=0 and x varies from 2 to 0.

Along the line from C(0,2) to O(0,0),x=0,dx=0 and y varies from 2 to 0.

LHS = \int _{{OA}}[(x^{2}-xy^{3})dx+(y^{2}-2xy)dy]+\int _{{AB}}[(x^{2}-xy^{3})dx+(y^{2}-2xy)dy]+\int _{{BC}}[(x^{2}-xy^{3})dx+(y^{2}-2xy)dy]+\int _{{CA}}[(x^{2}-xy^{3})dx+(y^{2}-2xy)dy]\,

=\int _{{x=0}}^{{2}}x^{2}dx+\int _{{y=0}}^{{2}}(y^{2}-4y)dy+\int _{{x=2}}^{{0}}(x^{2}-8x)dx+\int _{{y=2}}^{{0}}y^{2}dy\,

=[{\frac  {x^{3}}{3}}]_{0}^{2}+[{\frac  {y^{3}}{3}}-2y^{2}]_{0}^{2}+[{\frac  {x^{3}}{3}}-4x^{2}]_{2}^{0}+[{\frac  {y^{3}}{3}}]_{2}^{0}\,

={\frac  {8}{3}}+{\frac  {8}{3}}-8-{\frac  {8}{3}}=8\, --(3)

Hence,from (2) and (3), LHS=RHS, So Green's theorm is verified.

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