Calculus I · Unit 3B · lesson

Density and Mass

Concept

Learning objectives

Integrate linear density, area density, or volume density to find mass.

Density and Mass

Explanation

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 $dx$ has mass approximately $\lambda(x)dx$ , an area element has mass approximately $\rho(x)dA$ , and a volume element has mass approximately $\delta(x)dV$ . 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 $[a,b]$ with linear density $\lambda(x)$ mass per length,

m=\int_a^b\lambda(x)\,dx.

The same idea extends to surface and volume density with area or volume elements.

Application

A rod with increasing density

A 3-meter rod has density $\lambda(x)=2+x$ kilograms per meter. Its mass is

m=\int_0^3(2+x)dx=\frac{21}{2}\text{ kg}.

Interactive checku3b-mass-01

A 3-meter rod has density $2+x$ kg/m. Find its mass.

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Show hint

Integrate density over position.

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After the explanation

Use the section idea

Reading lens

Distinguish geometric size from weighted amount: arc and surface formulas stretch local distance, while density assigns unequal mass to equal pieces.

Mental model

Arc length adds short chord lengths; surface area adds narrow bands; mass adds density-times-size; moments add position-weighted mass.

Decision

Identify the local size element first, then multiply by circumference, density, or position only when the modeled quantity requires it.

Common trap

Using geometric midpoint for a nonuniform object, forgetting the surface radius, or treating differential legs as independent finite lengths.

Check yourself

Can you explain why every factor belongs in the slice contribution and why the result has the intended units?

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