Question

F(x,y) = $(x^2+xy$) ${\hat{a}}_x$ + $(y^2+xy$) ${\hat{a}}_y$. It's line integral over the straight line from (x,y) = (0,2) to (2,0) evaluates to

• A. -8
• B. 4
• C. 8
• D. 0

Answer 1

The correct option is D 0

Equation of straight line
$\frac{x}{2}+\frac{y}{2}$ = 1
$\Rightarrow$ x + y =2
Now, d$\overrightarrow{l}$ = $dx{\hat{a}}_x+dy{\hat{a}}_y$
and $\overrightarrow{F}=(x^2+xy){\hat{a}}_x+(y^2+xy){\hat{a}}_y$

Line integral of $\overrightarrow{F}$ (x,y)

${\int }_L\overrightarrow{F}(x,y).d\overrightarrow{l}={\int }_L(x^2+xy)dx+\left(\right(y^2+xy)dy$
$={\int }_0^2(x^2+x(2-x))dx+{\int }_0^2(y^2+(2-y)y)dy$
$={\left[x^2\right]}_0^2+{\left[y^2\right]}_2^0$
=4 - 0 + 0 - 4 = 0