Question

The line integral of the vector field $F=5xz\hat{i}+(3x^2+2y)\hat{j}+x^{2}z\hat{k}$ along a path form (0,0,0) to (1,1,1) parameterized by $(t,t^2,t)$ is

• A. 4.41

Answer 1

The correct option is A 4.41
$E=5xz\overline{i}+(3x^2+2y)\overline{j}+x^{2}z\overline{k}$
$={\int }_C\overline{F}.\overline{d}r$
$={\int }_{C}5xzdx+(3x^2+2y)dy+x^{2}zdz$
$x=t$, $y=t^2$, $z=t$, $t=0$ to 1
dx = dt
dy = 2t dt, dz = dt
$={\int }_0^{1}5t^{2}dt+(3t^2+2t^2)2tdt+t^{3}dt$
$={\int }_0^1(5t^2+11t^3)$dt
$={[\frac{5t^3}{3}+\frac{11t^4}{4}]}_0^1=\frac{5}{3}+\frac{11}{4}=\frac{53}{12}=4.41$