Two questions re. the calculation of total mass in a rod of non-uniform density

Question

I am currently learning about applying integration techniques to the calculation of mass in a rod of varying density. I feel as if I understand the general picture, but I have some specific points of confusion. Below I will explain the concept as I understand it, so any mistakes in my explanation can be pointed out:

Consider a straight rod of length $(b-a)$ made of a material with mass density that varies across the length of the rod.

We create a mass density function $\rho$, integrable on $[a,b]$, such that the mass density of the rod at a point, $x$, is given by $\rho (x)$. Recalling the definition of mass as the product of density and the space that density occupies, in this case, $\rho (x_k)$ and $dx$, the total mass of the rod, M, is given by the formula: $M=\int_{a}^{b}\rho (x) dx$

My two points of confusion are:

1. Couldn't mass density vary across the z-axis? And wouldn't this impact the total mass value of the rod? Does this mean we are implicitly assuming a constant mass density across the width of the rod?
2. When we evaluate a single $y$ value of the function at some point, what mass are we actually measuring? If $\rho (x_k)$ tells you the mass density at a point $x_k$ on the rod, and density is defined as the amount of mass per unit of space, what space in the rod are we talking about here? Are we calculating the amount of mass contained within a cross section of the rod of width $dx$ at the point $x_k$?

Answer 1

1. We use the case of a rod because it's (in theory) 1 dimensional, and thus we can neglect the varying mass density on the z axis. We do have tools and techniques for evaluating the same masses for surfaces and volumes, so if we had an equation that could modelise our rod in 3 dimensions and a mass density function to accompany it, we would then compute the surface or volume integrals of the function over the surface/volume (a great place to learn about these pieces of maths is in H.M Schey's Div, Grad, Curl and all that).

2. As just stated, in the rod example we consider it to be 1 dimensional. It only has a length, and no width or height. Evaluating a single $y$ value for $\rho(x)$ gives us the mass density at that point $x$ on the rod. If you prefer, you can imagine the rod literally being the interval $[a;b]$ and graph $\rho(x)$ directly above it, explicitly putting on display the variation of mass density in space. The infinite sum of the infinitesimal values of $\rho(x)$ along that interval of space consequentially yields the total mass of the rod, which is the literal definition of $\int_a^b\rho(x)\ dx$. This means that $\rho(x)$ only details how the mass is concentrated at some point on the rod $x$ in 1 dimension along the interval $[a;b]$. It is always best to have a concrete visualisation of what is going on.

Hopefully this clears things up a little!