A definite integral with a variable upper limit can create a new function that measures accumulated change. This idea links integration, area, and dynamic quantities in meaningful ways.

Defining Functions Using Definite Integrals

A core idea in AP Calculus AB is that definite integrals do more than compute single numerical values. When the upper limit becomes a variable, the integral defines a new function that measures accumulated change over an interval starting at a fixed point. This perspective forms the basis for the Fundamental Theorem of Calculus, which connects accumulation and differentiation in a powerful way.

Functions Defined by Integrals

Consider a continuous rate function $f(t)$. If we define a new function
$g(x) = \int_a^x f(t),dt$, then $g(x)$ represents the accumulated change of the quantity whose rate is $f$ from the starting value $a$ to the variable endpoint $x$.
This construction interprets integration dynamically: the output of $g(x)$ depends on how much area has accumulated under $f(t)$ as the upper limit moves.

This definition highlights that accumulation functions naturally encode the history of a rate over an interval, making them essential tools for modeling change.

Structure of an Accumulation Function

When interpreting or constructing a function like $G(x) = \int_a^x f(t),dt$, it is important to understand its components:

• Lower limit $a$: The fixed starting position or time.

• Variable upper limit $x$: The point to which accumulation is measured.

• Integrand $f(t)$: The rate of change of a quantity with respect to $t$.

• Output $G(x)$: The total change accumulated between $a$ and $x$.

Each of these elements contributes to the meaning and behavior of the accumulation function, and the structure ensures that the function is well defined whenever $f$ is continuous on the interval of interest.

Interpreting Accumulated Change as Area

Accumulation functions rely on a geometric interpretation: the accumulated change equals the signed area under the integrand from $a$ to $x$.
This area-based interpretation provides a visual way to understand how accumulation unfolds as the upper limit moves along the axis.

This diagram shows positive and negative shaded regions under the function $f(t)$ and a table displaying corresponding accumulation values. It emphasizes that areas above the axis contribute positively while areas below contribute negatively to net accumulated change. The table is an additional detail illustrating how specific accumulated values vary with different interval endpoints. Source.

Behavior as the Upper Limit Varies

As $x$ increases, the integral continuously gathers additional area under the graph of the rate function. This dynamic process means that an accumulation function typically contains rich information about the rate’s behavior over the interval. Key observations include:

• Increasing $x$ adds area, thereby increasing or decreasing the accumulated value depending on the sign of $f(t)$.

• If $f(t)=0$ over some interval, the accumulation remains constant on that interval.

• Sharp changes in $f(t)$ lead to correspondingly sharp changes in the accumulation function’s slope.

This understanding sets the stage for later applications of the Fundamental Theorem of Calculus, which formalizes how the derivative of an accumulation function relates to its integrand.

Why Definite-Integral Definitions Matter

Defining functions using definite integrals serves several essential purposes in calculus:

• It provides a general mechanism for constructing new functions from existing rate functions.

• It captures the cumulative nature of physical and mathematical processes, such as displacement from velocity or total growth from a rate of change.

• It links the geometric concept of area with the analytic concept of accumulated change.

• It prepares students to apply the Fundamental Theorem of Calculus to connect differentiation and integration.

Key Interpretation Principles

To analyze a function defined by an integral, use these principles:

• Focus on the rate function $f(t)$, since it determines the accumulation.

• Interpret the integral as tracking net area from $a$ to $x$.

• Recognize that the accumulation function measures change of a quantity, not the quantity itself unless additional information is provided.

• Relate any notable features of the integrand—such as zeros, maxima, or sign changes—to corresponding features in the accumulation function.

This image displays a shaded region under a positive function $f$ beginning at $x=a$, illustrating how a definite integral measures accumulated area. It visually reinforces the geometric basis of accumulation functions. The image is slightly more general because it does not label the upper limit explicitly, but the conceptual interpretation is identical. Source.