VC4.4

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Here,C is the straight line joining (0,0,0) and (1,1,1) whose equations are

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

Therefore,on C, x=t,y=t,z=t and so r=xi+yj+zk=ti+tj+tk,dr=(i+j+k)dt\,

Also,F=(3t^{2}+6t)i-14t^{2}j+20t^{3}k\,

The given points correspond to t=0 and t=1.

Therefore,\int _{{C}}F\cdot \,dr=\int _{{0}}^{{1}}[(3t^{2}+6t)i-14t^{2}j+20t^{3}k]\cdot (i+j+k)dt\,

=\int _{{0}}^{{1}}[(3t^{2}+6t)-14t^{2}+20t^{3}]\,dt=[t^{3}+3t^{2}-{\frac  {14t^{3}}{3}}+5t^{4}]_{{0}}^{{1}}={\frac  {13}{3}}\,

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