Calculus I · Unit 3B · lesson

Work and Variable Force

Concept

Learning objectives

Model work as the integral of force over displacement and interpret units.

Work and Variable Force

Explanation

Work accumulates force through distance

For a constant force acting in the direction of motion, work is force times distance. If the force varies with position, divide the displacement into short intervals. On each interval, the force is approximately constant, so the local work is $F(x_i^*)\Delta x$ . The integral $\int_a^b F(x)\,dx$ is the limiting total.

Signs and units carry physical meaning. Force in the direction of displacement contributes positive work, while an opposing force contributes negative work. In SI units, newtons times meters give joules. A graph of force versus position makes the interpretation visible: work is signed area under the force curve, not area under a position or time graph unless the variables have been transformed appropriately.

For constant force parallel to motion, $W=Fd$ . For varying force,

W=\int_a^bF(x)\,dx.

In SI units, newtons times meters gives joules.

Application

A linearly increasing force

A force $F(x)=10+3x$ newtons moves an object from $x=0$ to $x=4$ meters:

W=\int_0^4(10+3x)dx=64\text{ J}.

Interactive checku3b-work-01

A force $F(x)=10+3x$ N moves an object from 0 to 4 m. Find the work.

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Show hint

Integrate force with respect to displacement.

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**Visual study**
Work as force-distance accumulation. Show force curve and thin work contribution F(x)dx.

After the explanation

Use the section idea

Reading lens

Translate the situation into rate, density, force, pressure, or probability before calculating; the integral is the final accumulation step, not the first modeling decision.

Mental model

Work adds force through distance, pumping adds slice weight through lift distance, pressure adds depth-dependent strip force, and marginal or probability models add weighted local contributions.

Decision

Draw a coordinate system, define the slice at a general position, express every changing factor in one variable, and state the domain and units.

Common trap

Confusing mass density with weight density, measuring depth from the wrong reference, or integrating a marginal quantity without an initial value when a total function is requested.

Check yourself

Does the setup respond correctly when the slice moves, and can a units or scale check expose a missing factor?

Source & rights

Original instruction with traceable references.

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