= Vector line integral weighted by a scalar field
{title2=$\int_C V\,d\mathbf r$}
For an oriented <parametric curve> $\mathbf r(t)$ and scalar field $V$, define the componentwise <line integral>
$$
\int_C V\,d\mathbf r=\int V(\mathbf r(t))\mathbf r'(t)\,dt.
$$
Its value is a <vector> and reversing orientation changes its sign. It differs from the scalar <arc length> integral $\int_CV\,ds=\int V(\mathbf r(t))|\mathbf r'(t)|\,dt$, which is independent of orientation. A request to integrate a scalar field along a curve should specify which differential is intended.
For the counterclockwise circle $\mathbf r(\phi)=(R\cos\phi,R\sin\phi,z_0)$, suppose the restricted field has <Fourier series> $V(\phi)=A_0+\sum_{n\geq1}(A_n\cos n\phi+B_n\sin n\phi)$. <Orthogonality> gives
$$
\oint_C V\,d\mathbf r=\pi R(-B_1,A_1,0),\qquad
\oint_C V\,ds=2\pi RA_0.
$$
The oriented vector integral selects the first angular harmonics, whereas the scalar arc-length integral selects the mean.
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