I'm trying to solve
$\int\left [ -KR^{2} \right ]\hat{z}d\vec{l}\cdot \hat{a}$
around a circular loop which popped up in a physics working.
However, I am unable to do so.
To kick start the attempt, I understand that the integration limits involves only the polar angle $\phi$ over the interval from 0 to 2$\pi$. The cylindrical radius s=R is held fixed as we integrate over the loop. However, I am unable to break down d$\vec{l}$ into its constituent components in cylindrical coordinates.
Since the loop is a circle, say of radius $R$, we only care about the contribution in the direction of $\hat \phi$. Suppose we are at a point where the angle with the positive $x$-axis is $\phi$. We have that $$ d\vec\ell = R\,d\phi\, \hat \phi. $$ Geometrically, this approximates the small portion of arc length at the angle $\phi$, given generally as $R\Delta \phi\to R\,d\phi$ as an infinitesimal.