In physical situations, you may want to add up something which is not constant from place to place. To do this approximately, it is simple to imagine chopping the relevant part of space up into little pieces, small enough that the something is essentially constant on each little piece and then adding up the contribution from each little piece. Mathematicians spent 200 years understanding what it means to chop the space up into pieces that are “infinitesimally small” and showing that, in this limiting case, the sum becomes an integral. The beauty of their results are that, in all cases relevant to us, the naive computation works perfectly; we are now free to take advantage of their hard work without further ado.

Sometimes you want to find the total amount of some scalar quantity (like mass or charge). If the stuff is spread out on a one-dimensional curve (like a wire), then you need to chop the curve into a lot of little pieces each of which is a little length (usually called $ds$, where $s$ is the parameter that labels position on the wire, and the $d$ indicates an infinitesimally small piece of length, technically called a differential). Most often, the information that you have is the amount of stuff per unit length, which may be either a constant or a function that depends on the position on the wire, (e.g. $\hbox{kg/m}$). This quantity is called a linear density and is usually denoted ($\lambda(s)$, where $\lambda$ is the greek letter “l”, for length, and the notation shows explicitly that the density varies from point to point on the wire, i.e. it is a function of $s$). Then the quantity $\lambda(s)\, ds$ is the amount of stuff on the little length $ds$ and the total amount of stuff ($M$, for mass, for example) is given by: \begin{equation} M=\int\limits_{\hbox{wire}} \lambda(s)\, ds \end{equation}

If the stuff is spread out on a two-dimensional surface, not necessarily flat, then you need to chop the surface into a lot of little pieces each of which is a little area (variously called $da$, $dA$, $dS$, or $d\sigma$, where $\sigma$ is the greek letter “s”, for surface).