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GATE 2016 EE - SET 2 -Question 31

F=5xz\hat{i}+(3x^{2}+2y)\hat{j}+x^{2}z\hat{k}

x=t, y=t^{2}, z=t\Rightarrow dx=dt, dy=2tdt, dz=dt

\int_{(0,0,0)}^{(1,1,1)}\left [ 5xzdx+(3x^{2}+2y)dy+x^{2}zdz \right ]

\Rightarrow \int_{t=0}^{1}\left [ 5(t)(t)dt+(3t^{2}+2t^{2})2tdt+(t^{2})(t)dt \right ]

\Rightarrow \int_{t=0}^{1}\left [ 5t^{2}dt+10t^{3}dt+t^{3}dt \right ]

\Rightarrow \int_{0}^{1}\left [ 5t^{2}+11t^{3} \right ]dt\Rightarrow \left [ \frac{5t^{3}}{3}+\frac{11t^{4}}{4} \right ]_{0}^{1}=\frac{53}{12}=4.416

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