VC4.16

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Let C be the given curve from t=1 to t=2. The parametric equations of the curve are x=t^{2}+1,y=2t^{2},z=t^{3}\, --(1)

Hence,the required total workdone

\int _{C}F\cdot \,dr=\int _{C}(3xyi-5zj+10xk)\cdot (dxi+dyj+dzk)\,

=\int _{C}(3xy{\frac  {dx}{dt}}-5z{\frac  {dy}{dt}}+10x{\frac  {dz}{dt}}\,dt\,

=\int _{1}^{2}[3(t^{2}+1)(2t^{2})(2t)-(5t^{3})(4t)+10(t^{2}+1)(3t^{2})\,dt\, using (1)

=\int _{1}^{2}(12t^{5}+10t^{4}+12t^{3}+30t^{2})\,dt=[2t^{6}+2t^{5}+2t^{4}+10t^{3}]_{1}^{2}\,

=2(2^{6}-1)+2(2^{5}-1)+2(2^{4}-1)+10(2^{3}-1)=303\,

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