Question

$\text{Given a vector field}\overrightarrow{F}=yx{\hat{a}}_x-y^{2}z{\hat{a}}_y-2x^2{\hat{a}}_z,\text{the line integral}\int \overrightarrow{F}.\overrightarrow{dl}\text{evaluated along a segment}\text{on the x-axis from}x=1\text{to}x=3\text{is}$

• A. 0
• B. 1.66
• C. 3.33
• D. -3.33

Answer 1

The correct option is A 0

To find $\int \overrightarrow{F}.\overrightarrow{dt}$ along a segment on the x-axis from x = 1 to x = 3.
i.e. y = 0, z = 0, dy = 0 and dz = 0
$\int \overrightarrow{F}.\overrightarrow{dl}=\int (yx\widehat{a_x}-y^{2}z\widehat{a_y}-2x^2\widehat{a_z}).(\widehat{a_x}dx+\widehat{a_y}dy+\widehat{a_z}dz)$
$=\int (yxdx-y^{2}zdy-2x^{2}dz)$

Putting, y = 0 , z = 0, dy = 0 and dz = 0
we get, $\int \overrightarrow{F}.\overrightarrow{dl}=0$