Calculus I · Unit 3A · lesson

The Net Change Theorem

Concept

Learning objectives

Translate a contextual rate integral into final amount minus initial amount.

The Net Change Theorem

Explanation

Rates become measurable changes

If $Q'(t)$ is the rate at which a quantity changes, then $\int_a^b Q'(t)\,dt=Q(b)-Q(a)$ . This net-change statement is the Fundamental Theorem written in applied language. It tells us that integrating a rate over time recovers the actual change in the underlying quantity, whether that quantity is position, volume, charge, population, energy, or cost.

The word "net" matters whenever the rate can be negative. Inflow and outflow, charging and discharging, or forward and backward motion contribute with opposite signs. To recover the final amount, add the net change to the initial amount. To recover total activity without cancellation, a different model may be needed, often involving absolute value or separate positive and negative contributions.

If $Q'(t)$ is a rate of change, then

\int_a^bQ'(t)\,dt=Q(b)-Q(a).

This is FTC Part II interpreted in context.

Application

Charge flowing through a circuit

Current $I(t)$ , measured in amperes, is charge flow in coulombs per second. If

I(t)=3e^{-t}+1

for $0\le t\le4$ , then the net charge transferred is

\int_0^4I(t)\,dt=[-3e^{-t}+t]_0^4=7-3e^{-4}\text{ coulombs}.

The units make the interpretation unavoidable: amperes times seconds equals coulombs.

Interactive checku3a-net-change-01

A population changes at rate $P'(t)=3t$ people/year for $0\le t\le4$ . What is the net population change?

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Show hint

Integrate the rate over the time interval.

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After the explanation

Use the section idea

Reading lens

Use the Fundamental Theorem as the bridge between accumulation functions, local rates, and endpoint evaluation.

Mental model

Differentiating a running total recovers its current integrand, while evaluating an accumulated total subtracts antiderivative endpoint values.

Decision

Separate FTC Part I, FTC Part II, net change, and variable-bound chain-rule tasks before manipulating notation.

Common trap

Forgetting a chain-rule factor at a variable bound, reversing endpoint subtraction, or adding +C to a definite value.

Check yourself

Can you state which part of the theorem applies and why its hypotheses and bounds fit?

Source & rights

Original instruction with traceable references.

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