GATE 2016 EE – SET 2 -Question 31

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

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

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

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

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

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