VC5.5

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Note thatsemi-circle y=(1-x^{2})^{{{\frac  {1}{2}}}}\, is lying above the x-axis.

Thus,C consists of part say AB of the x-axis and the semi-circle BDA(say). Let R be the plane region bounded by C.Using Green's theorm, we have

\int _{C}[(2x^{2}-y^{2})dx+(x^{2}+y^{2})dy]=\iint _{R}[{\frac  {\partial }{\partial }}(x^{2}+y^{2})-{\frac  {\partial }{\partial y}}(2x^{2}-y^{2})]dxdy=\iint _{R}(2x+2y)dxdy\,

=2\int _{{x=-1}}^{{1}}\int _{{y=0}}^{{{\sqrt  {(1-x^{2})}}}}(x+y)dxdy\, [Noting that for the region R,y varies from 0 to {\sqrt  {(1-x^{2})}}\, and x varies from -1 to 1].

=2\int _{{x=-1}}^{{1}}[xy+{\frac  {y^{2}}{2}}]_{{0}}^{{{\sqrt  {(1-x^{2})}}}}dx\, on integrating w.r.t.y

=2\int _{{-1}}^{{1}}[x{\sqrt  {1-x^{2}}}+{\frac  {1}{2}}(1-x^{2})]dx=0+2\int _{0}^{1}(1-x^{2})dx\,

=2[x-{\frac  {x^{3}}{3}}]_{0}^{1}=2[1-{\frac  {1}{3}}]={\frac  {4}{3}}\,

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