Mass and Density - One Dimenstion

Consider a one-dimensional object such as a length of wire. We ignore the atomic nature of matter and assume that it is distributed continuously along a line segment. If the density ρ per unit length is the same at each point of the wire, then the mass is the product of the density and the length, m = ρL. If L is in centimeters and ρ in grams per centimeter, then m is in grams, (ρ is the Greek letter "rho".)

Now suppose that the density of the wire varies continuously with the position. Put the wire on the x-axis between the points x = a and x = b, and let the density at the point x be ρ(x). Consider the piece of the wire of infinitesimal length Δx and mass Δm shown in Figure 6.6.1. At each point between x and x + Δx, the density is infinitely close to ρ(x), so

Δm ≈ ρ(x) Δx (compared to Δx).

Figure 6.6.1

Therefore by the Infinite Sum Theorem, the total mass is

Example 1