Area density

Definition

Area density, also known as surface density, is a measure of mass per unit area. It is typically denoted by the symbol $\sigma$ and has units of $\text{kg}/\text{m}^2$.

5 Must Know Facts For Your Next Test

1. Area density is calculated as $\sigma = \frac{dm}{dA}$, where $dm$ is the infinitesimal mass element and $dA$ is the infinitesimal area element.
2. In applications of integration, area density can be integrated over a surface to find total mass: $M = \int_A \sigma \, dA$.
3. Area density problems often involve setting up integrals in polar or Cartesian coordinates depending on symmetry.
4. Understanding transformations between coordinate systems (e.g., Cartesian to polar) is crucial when working with area densities on non-rectangular surfaces.
5. The concept is important in physical applications such as calculating the mass of thin plates or membranes.

Review Questions

• How do you express area density mathematically?
• What integral would you set up to find the total mass of an object given its area density function?
• Explain how you would convert an area integral from Cartesian coordinates to polar coordinates for a circular region.

Related terms

Volume Density: Volume density (or mass density) measures the amount of mass per unit volume and is denoted by $\rho$. Units are typically $\text{kg}/\text{m}^3$.

Linear Density:

Linear density measures mass per unit length and is represented by $\lambda$. Units are usually $\text{kg}/\text{m}$.

Moment of Inertia:

$I = \int r^2 \, dm$, where \(r\) is the distance from the axis. It quantifies an object's resistance to rotational motion.