VC4.18

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Let C denote the straight line joining (0,0,0) to (2,1,3). Then the equations of the line C are

{\frac  {x-0}{2-0}}={\frac  {y-0}{1-0}}={\frac  {z-0}{3-0}}=t\, (say)

Hence the parametric equations of line C are x=2t,y=t,z=3t\, --(1)

At the point (0,0,0),t=0 and at the point (2,1,3),t=1. Since r=xi+yj+zk,dr=dxi+dyj+dzk\, Therefore,the required work done is \int _{C}F\cdot \,dr=\int _{C}[3x^{2}i+(2xz-y)j+zk]\cdot (dxi+dyj+dzk)\,

=\int _{C}[3x^{2}dx+(2xz-y)dy+zdz]=\int _{{t=0}}^{{1}}[3x^{2}{\frac  {dx}{dt}}+(2xz-y){\frac  {dy}{dt}}+z{\frac  {dz}{dt}}]\,dt\,

=\int _{0}^{1}[(12t^{2})(2)dt+(12t^{2}-t)(1)+(3t)(3)dt\,

=\int _{0}^{1}(36t^{2}+8t)dt=36[{\frac  {1}{3}}t^{3}]_{0}^{1}+8[{\frac  {1}{2}}t^{2}]_{0}^{1}=12+4=16\,


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