Integration is fundamentally about adding up small quantities, but there is a conceptual difficulty here which is often overlooked. It's one thing to say that the mass of a wire is given by $m=\int dm=\int\rho\,ds$, and quite another to say that the mass of a ball is given by $m=\int dm=\int\rho\,dV$. In the former integral, “$d$” can be interpreted as “small in one direction”, but in the latter it means “small in three directions”.

A common student error arises from this somewhat inconsistent notation: Since the volume of a sphere is given by $V=\frac43\pi r^3$, surely the volume element must be given by $dV=4\pi r^2\,dr$. It doesn't help that this argument is in fact correct for problems with spherical symmetry, which forms the basis of the shell method for volume integrals. The need to keep track of what “small” means — and the use of the same symbols in multiple contexts — are key ideas in multivariable calculus which are not always discussed explicitly.