Calculus I · Unit 3B · lesson
Density and Mass
Learn Density and Mass through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Length, surface, mass, and balanceWhat this section is building
Learn Density and Mass through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Arc length adds short chord lengths; surface area adds narrow bands; mass adds density-times-size; moments add position-weighted mass.
Identify the local size element first, then multiply by circumference, density, or position only when the modeled quantity requires it.
Using geometric midpoint for a nonuniform object, forgetting the surface radius, or treating differential legs as independent finite lengths.
Learning objectives
Integrate linear density, area density, or volume density to find mass.
Density and Mass
Density tells how mass is distributed
When density is constant, mass is density times size. When density varies, divide the object into small pieces. A short rod segment of length has mass approximately , an area element has mass approximately , and a volume element has mass approximately . Integration adds the nonuniform contributions.
The form of the density must match the dimension of the object. Linear density has units of mass per length, surface density has mass per area, and volume density has mass per volume. Multiplying by the corresponding differential element should leave units of mass. This unit cancellation is one of the cleanest ways to verify that the model uses the correct density and integration variable.
For a thin rod along with linear density mass per length,
The same idea extends to surface and volume density with area or volume elements.
A rod with increasing density
A 3-meter rod has density kilograms per meter. Its mass is
u3b-mass-01A 3-meter rod has density kg/m. Find its mass.
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Integrate density over position.
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