Section 7.2 Scalar Line Integrals
What if you want to determine the mass of a wire in the shape of the curve \(C\) if you know the density \(\lambda\text{?}\) The same procedure still works; chop and add. In this case, the length of a small piece of the wire is \(ds=|d\rr|\text{,}\) so its mass is \(\lambda\,ds\text{,}\) and the integral becomes
which can also be written as
which emphasizes both that \(\lambda\) is not constant, and that \(ds\) is the magnitude of \(d\rr\text{.}\)
Another standard application of this type of line integral is to find the center of mass of a wire. This is done by averaging the values of the coordinates, weighted by the density \(\lambda\) as follows:
with \(m\) as defined above. Similar formulas hold for \(\bar{y}\) and \(\bar{z}\text{;}\) the center of mass is then the point \((\bar{x},\bar{y},\bar{z})\text{.}\)