What is the difference between the line integrals $\oint_b\,ds$, $\oint_b\,dx$, and $\oint_b\,x\,ds$? Can someone explain to me what this means: Where c is the path of a circle with some radius b in clockwise direction. $\oint_b \,ds$ $\oint_b \,dx$ $\oint_b \,xds$ Like I can't visualise this. Can someone help me?

* The first integral is the arclength of the curve. (Since your curve is a circle of radius $b$, this is just $2\pi b$.)
* The second integral is the line integral of the vector field $\langle x,y\rangle=\langle 1,0\rangle$ along the curve. By symmetry, this is zero.
* The third integral is the "mass" of the curve if the "mass" density per unit length is $x$. (Of course, $x$ won't really be a _mass_ density, because it takes negative values. The point is that, in general, we interpret the integral $\int_c f\,ds$ as the integration of the density of some quantity per unit length.) Again by symmetry, the integral is zero.