VC4.11

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Along the given curve C,we have r=xi+yj+zk=2t^{2}i+tj+t^{3}k\,

Also,A=(2t+3)i+2t^{5}j+(t^{4}-2t^{2})k\, [On putting the values of x,y,z]

From the first equation,{\frac  {dr}{dt}}=4ti+j+3t^{2}k\,

Therefore,\int _{{C}}A\cdot \,dr=\int _{{C}}[A\cdot {{\frac  {dr}{dt}}}]\,dt\,

=\int _{{0}}^{{1}}[(2t+3)i+2t^{5}j+(t^{4}-2t^{2})k]\cdot (4ti+j+3t^{2}k)\,dt\,

=\int _{{0}}^{{1}}[4t(2t+3)+2t^{5}+3t^{2}(t^{4}-2t^{2})dt\,

=\int _{{0}}^{{1}}(8t^{2}+12t+2t^{5}+3t^{6}-6t^{4})\,dt\,

=[{\frac  {8t^{3}}{3}}+6t^{2}+{\frac  {t^{6}}{3}}+{\frac  {3t^{7}}{7}}-{\frac  {6t^{5}}{5}}]_{{0}}^{{1}}\,

={\frac  {8}{3}}+6+{\frac  {1}{3}}+{\frac  {3}{7}}-{\frac  {6}{5}}={\frac  {864}{105}}={\frac  {288}{35}}\,


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