Calculus I · Unit 3B · lesson

Recovering Totals from Marginal Quantities

Concept

Learning objectives

Integrate marginal cost, revenue, or profit to recover total change.

Recovering Totals from Marginal Quantities

Explanation

A marginal function is a rate of total change

In economics, a marginal quantity is a derivative. Marginal cost $C'(q)$ measures the approximate change in total cost for one additional unit near production level $q$ . Integrating a marginal function over a production interval recovers the net change in the corresponding total function. One known total value is then needed to recover the absolute total.

The interpretation should not be reduced to symbol manipulation. The integral $\int_a^b C'(q)\,dq$ is the increase in cost from producing $a$ units to producing $b$ units, not the total cost at $b$ unless $C(a)=0$ . Units provide the distinction: dollars per unit times units gives dollars. The same reasoning applies to marginal revenue, marginal profit, and other accumulated business quantities.

If $C'(q)$ is marginal cost in dollars per unit, then the added cost of increasing production from $q=a$ to $q=b$ is

C(b)-C(a)=\int_a^bC'(q)dq.

An initial fixed cost determines the constant when a complete cost function is required.

Application

Production expansion

Suppose $C'(q)=20+0.04q$ dollars per item and $C(0)=5000$ . Then

C(q)=20q+0.02q^2+5000.

The cost of expanding production from 100 to 150 items is

\int_{100}^{150}(20+0.04q)dq.

Interactive checku3b-marginal-01

If $C'(q)=20+0.04q$ , find the added cost from $q=100$ to $q=200$ .

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Show hint

Integrate marginal cost over the production interval.

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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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