Physical Applications: Learn It 1

  • Calculate the mass of linear and circular objects using their density distributions
  • Compute the work required in various situations, such as pumping fluids or moving objects along a path
  • Determine the force exerted by water on a vertical surface underwater

Mass and Density

Mass–Density Formula of a One-Dimensional Object

We can use integration to calculate the mass of a thin rod based on a density function. Let’s consider a rod oriented along the [latex]x[/latex]-axis from [latex]x=a[/latex] to [latex]x=b[/latex] (Figure 1). 

This figure has the x and y axes. On the x-axis is a cylinder, beginning at x=a and ending at x=b.
Figure 1. We can calculate the mass of a thin rod oriented along the [latex]x\text{-axis}[/latex] by integrating its density function.

Note that although we depict the rod with some thickness in the figures, for mathematical purposes we assume the rod is thin enough to be treated as a one-dimensional object.

If the rod has constant density [latex]\rho ,[/latex] given in terms of mass per unit length, then the mass of the rod is just the product of the density and the length of the rod: [latex](b-a)\rho .[/latex]

If the density varies along the rod, we use a linear density function [latex]\rho (x)[/latex]. Let [latex]\rho (x)[/latex] be an integrable linear density function. Partition the interval [latex][a,b][/latex] into [latex]n[/latex] segments, each of width Δx.

This figure has the x and y axes. On the x-axis is a cylinder, beginning at x=a and ending at x=b. The cylinder has been divided into segments. One segment in the middle begins at xsub(i-1) and ends at xsubi.
Figure 2. A representative segment of the rod.

The mass of a segment [latex][x_{i−1},x_{i}][/latex] is:

[latex]{m}_{i}=\rho ({x}_{i}^{*})\text{Δ}x.[/latex]

Summing these segments gives an approximation for the total mass:

[latex]m=\displaystyle\sum_{i=1}^{n} {m}_{i}\approx \displaystyle\sum_{i=1}^{n} \rho ({x}_{i}^{*})\text{Δ}x.[/latex]

This is a Riemann sum. Taking the limit as [latex]n\to \infty ,[/latex] we get an expression for the exact mass of the rod:

[latex]m=\underset{n\to \infty }{\text{lim}}\displaystyle\sum_{i=1}^{n} \rho ({x}_{i}^{*})\text{Δ}x={\displaystyle\int }_{a}^{b}\rho (x)dx.[/latex]

We state this result in the following theorem.

mass–density formula of a one-dimensional object

Given a thin rod oriented along the [latex]x\text{-axis}[/latex] over the interval [latex]\left[a,b\right],[/latex] let [latex]\rho (x)[/latex] denote a linear density function giving the density of the rod at a point [latex]x[/latex] in the interval. Then the mass of the rod is given by

[latex]m={\displaystyle\int }_{a}^{b}\rho (x)dx[/latex]

Consider a thin rod oriented on the [latex]x[/latex]-axis over the interval [latex]\left[\frac{\pi}{2},\pi \right].[/latex] If the density of the rod is given by [latex]\rho (x)= \sin x,[/latex] what is the mass of the rod?