# Mass and Density

## Mass of a Thin Rod

We can use integration for calculating mass based on a density function.

Consider a thin wire or rod that is located on an interval [*a*, *b*].

The density of the rod at any point *x* is defined by the density function *ρ* (*x*). Assuming that *ρ* (*x*) is an integrable function, the mass of the rod is given by the integral

## Mass of a Thin Disk

Suppose that \(\rho \left( r \right)\) represents the radial density of a thin disk of radius \(R.\)

Then the mass of the disk is given by

## Mass of a Region Bounded by Two Curves

Suppose a region is enclosed by two curves \(y = f\left( x \right),\) \(y = g\left( x \right)\) and by two vertical lines \(x = a\) and \(x = b.\)

If the density of the lamina which occupies the region only depends on the \(x-\)coordinate, the total mass of the lamina is given by the integral

where \(f\left( x \right) \ge g\left( x \right)\) on the interval \(\left[ {a,b} \right],\) and \({\rho \left( x \right)}\) is the density of the material changing along the \(x-\)axis.

## Mass of a Solid with One-Dimensional Density Function

Consider a solid \(S\) that extends in the \(x-\)direction from \(x = a\) to \(x = b\) with cross sectional area \(A\left( x \right).\)

Suppose that the density function \(\rho \left( x \right)\) depends on \(x\) but is constant inside each cross section \(A\left( x \right).\)

The mass of the solid is

## Mass of a Solid of Revolution

Let \(S\) be a solid of revolution obtained by rotating the region under the curve \(y = f\left( x \right)\) on the interval \(\left[ {a,b} \right]\) around the \(x-\)axis.

If \(\rho \left( x \right)\) is the density of the solid material depending on the \(x-\)coordinate, then the mass of the solid can be calculated by the formula

## Solved Problems

### Example 1.

A rod with a linear density given by \[\rho \left( x \right) = {x^3} + x\] lies on the \(x-\)axis between \(x = 0\) and \(x = 2.\) Find the mass of the rod.

Solution.

We need to integrate the following:

If \(\rho\) is measured in kilograms per meter and \(x\) is measured in meters, then the mass is \(m = 6\,\text{kg}.\)

### Example 2.

Let a thin rod of length \(L = 10\,\text{cm}\) have its mass distributed according to the density function \[\rho \left( x \right) = 50{e^{ - \frac{x}{{10}}}},\] where \(\rho \left( x \right)\) is measured in \(\frac{\text{g}}{\text{cm}},\) \(x\) is measured in \(\text{cm}.\) Calculate the total mass of the rod.

Solution.

To find the mass of the rod we integrate the density function:

### Example 3.

Suppose that the density of cars in traffic congestion on a highway changes linearly from 30 to 150 cars per km per lane on a \(5\,\text{km}\) long stretch. Estimate the total number of cars on the highway stretch if it has \(4\) lanes.

Solution.

First we derive the equation for the density function \(\rho \left( x \right).\) Since the function is linear, it is defined by two points:

Using the two-point form of a straight line equation, we have

Now, to estimate the amount of cars on the highway stretch, we integrate the density function and multiply the result by \(4:\)

### Example 4.

Determine the total amount of bacteria in a circular petri dish of radius \(R\) if the density at the center is \({\rho_0}\) and decreases linearly to zero at the edge of the dish.

Solution.

The density of bacteria varies according to the law

where \(0 \le r \le R.\)

To find the total number of bacteria in the dish, we use the formula

This yields: