How is the differential mass element derived from a variable density function?

Conditions

• The object has non-uniform density.
• The density is modeled as a function of position $(x,y,z)$.
• $dV$ denotes an infinitesimal volume element.

Reasoning, step by step

1. Identify the density function $\rho(x,y,z)$.
2. Identify the infinitesimal volume element $dV$.
3. Multiply the density by the volume element to get the mass of the small piece.
4. Write the result as $dmass = \rho(x,y,z) \, dV$.

Example

The speaker says the mass of any small volume would be that volume times the density, and introduces the notation $dmass$. The board shows $dmass = xyz \, dV$.

Common misconceptions

• Using total volume times a single density value when density varies.
• Confusing the differential mass element with the total mass.

Watch the explanation

Explore next

• What does $\rho(x,y,z)=xyz$ mean in this triple-integral lesson?
• Is $dV$ always equal to $dx \, dy \, dz$ in triple integrals?