Question

* The line integral of function F=yzi, in the counter clockwise direction, along the circle $x^2+y^2=1$ at z = 1 is

• A. -2$\pi$
• B. -$\pi$
• C. $\pi$
• D. 2$\pi$

Answer 1

The correct option is B -$\pi$

C: $x^2+y^2=1$, z = 1
$\overrightarrow{F}=yz\hat{i},d\overrightarrow{r}=dx\hat{i}+dy\hat{j}+dz\overrightarrow{k}$
$x=cos\theta ,y=sin\theta ,\theta ϵ[0,2\pi ]$
Line integral $={\int }_c\overrightarrow{F}.d\overrightarrow{r}$
$={\int }_{c}yzdx$
$={\int }_0^{2\pi }sin\theta .1(-sin\theta )d\theta$
$=-{\int }_0^{2\pi }\frac{1-cos2\theta }{2}d\theta$
$=-\frac{1}{2}{[\theta -\frac{sin2\theta }{2}]}_0^{2\pi }=-\pi$