Density function

Definition

A density function in calculus represents the distribution of mass or probability over a given interval. It is often used to calculate properties like total mass, center of mass, and moments through integration.

5 Must Know Facts For Your Next Test

1. The integral of a density function over an interval gives the total mass or probability for that interval.
2. A continuous density function $\rho(x)$ describes how mass is distributed along a rod or other object.
3. The center of mass can be found using the formula $\frac{1}{M} \int_a^b x \rho(x) \, dx$, where $M$ is the total mass.
4. For probability density functions, the area under the curve must equal 1 over the entire range.
5. Moments about a point can be calculated using integrals involving the density function and distance terms.

Review Questions

• What is the integral of a density function over an interval used to find?
• How would you find the center of mass for a rod with a given density function $\rho(x)$?
• Why must the area under a probability density function equal 1?

Related terms

Center of Mass: The point at which the entire mass of an object can be considered to be concentrated for purposes of analysis.

Moment: A measure of the distribution of mass relative to a point or axis, often calculated as an integral involving distance terms.

$\int_a^b \rho(x) \, dx$: The integral used to calculate total mass or probability from a density function $\rho(x)$ over an interval [a, b].