VC4.15

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Given curve is given by x=t^{2},y=2t,z=t^{3}\, --(1)

Also,r=xi+yj+zk,dr=dxi+dyj+dzk\, --(2)

F\times dr={\begin{vmatrix}i&j&k\\xy&-z&x^{2}\\dx&dy&dz\end{vmatrix}}\,

=-(zdz+x^{2}dy)i-(xydz-x^{2}dx)j+(xydy+zdx)k\,

=-[t^{3}(3t^{2}dt)+t^{4}(2dt)]i-[2t^{3}(3t^{2}dt)-t^{4}(2tdt)]j+[2t^{3}(2dt)+t^{3}(2tdt)]k\,

=[-(3t^{5}+2t^{4})i-4t^{5}j+(4t^{3}+2t^{4})k]dt\,

Therefore,\int _{C}F\times dr=-i\int _{0}^{1}(3t^{5}+2t^{4})dt-4j\int _{0}^{1}t^{5}dt+2k\int _{0}^{1}(2t^{3}+t^{4})dt\,

=-i[3{\frac  {t^{6}}{6}}+2{\frac  {t^{5}}{5}}]_{0}^{1}-4j[{\frac  {t^{6}}{6}}]_{0}^{1}+2k[2{\frac  {t^{4}}{4}}+{\frac  {t^{5}}{5}}]_{0}^{1}\,

=-({\frac  {9}{10}})i-{\frac  {2}{3}}j+{\frac  {7}{5}}k\,

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