VC5.15

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Here the path of integration C consists of the straight lines OA,AB and BO where the coordinates of A and B are (2,0) and (2,1) respectively.Let R be the plane region bounded by C.

\oint _{C}F\cdot dr=\oint _{C}[(2x^{2}+y^{2})i+(3y-4x)j]\cdot (dxi+dyj)\,

=\oint _{C}[(2x^{2}+y^{2})dx+(3y-4x)dy]\,

=\iint _{R}[{\frac  {\partial }{\partial x}}(3y-4x)-{\frac  {\partial }{\partial y}}(2x^{2}+y^{2})]dxdy\,

=\iint _{R}(-4-2y)dxdy=-2\int _{{x=0}}^{{2}}\int _{{y=0}}^{{{\frac  {x}{2}}}}(2x+y)dxdy\,

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

=-2\int _{0}^{2}[x+{\frac  {x^{2}}{8}}]dx=-2[{\frac  {x^{2}}{2}}+{\frac  {x^{3}}{24}}]_{0}^{2}\,

=-2[2+{\frac  {1}{3}}]=-{\frac  {14}{3}}\,

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