Calculus I · Unit 3A · lesson

Accumulation Functions

Concept

Learning objectives

Interpret a variable-upper-limit integral as a new function and reason about its increase, decrease, and units.

Accumulation Functions

Explanation

An integral can define a new function

When the upper limit varies, $A(x)=\int_a^x f(t)\,dt$ records how much signed accumulation has occurred by the time the input reaches $x$ . The dummy variable $t$ is used inside the integral so that $x$ can remain free to control the endpoint. As $x$ moves, the accumulated region grows, shrinks, or gains negative contribution depending on the sign of $f$ .

The graph of $A$ can often be predicted directly from the graph of $f$ . Where $f$ is positive, $A$ increases; where $f$ is negative, $A$ decreases; where $f=0$ , $A$ has a horizontal tangent. This makes accumulation functions a bridge between geometric area, net change, and derivative behavior, setting up the Fundamental Theorem rather than appearing as an isolated notation trick.

Fix a starting point $a$ and define

A(x)=\int_a^x f(t)\,dt.

The variable $t$ is a dummy variable inside the integral; $x$ determines where accumulation stops.

If $f>0$ , moving $x$ to the right adds positive contributions and $A$ increases. If $f<0$ , $A$ decreases. The value $A(x)$ is a total; the value $f(x)$ is the local rate at which that total changes.

Worked example

Build an accumulation function directly

For $f(t)=2t$ and $a=1$ ,

A(x)=\int_1^x2t\,dt=x^2-1.

Thus $A(1)=0$ , $A(2)=3$ , and $A'(x)=2x=f(x)$ .

Interactive checku3a-accumulation-01

For $A(x)=\int_3^x f(t)\,dt$ , what is $A(3)$ ?

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**Visual study**
Moving upper bound and accumulated area. Show A(x)=int a x f(t)dt changing as x moves.

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?

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