AP Calculus AB Unit 6: Integration and Accumulation of Change Notes

Introduction

Integration adds refined signed changes. Riemann sums explain the construction; definite-integral notation records the limit.

The Fundamental Theorem connects integrands, accumulation functions, and antiderivative endpoint differences.

Symbolic methods are chosen from structure and verified by differentiation.

Current exam guidance: May 2027 format: 42 multiple-choice questions and 6 free-response questions. Calculator availability changes by part; the reader marks calculator decisions separately from mathematical justification.

6.1

Exploring Accumulations of Change

How does a rate determine net change and final amount?

Short answer

Add rate times input width across the interval; the limiting signed sum is a definite integral.

Know firstRate units · Signed area · Finite sums

Add small changes with their signs

Over a short interval, change is approximately rate times width. Adding these products and refining the intervals produces the definite integral. Above-axis rate adds and below-axis rate subtracts.

Net change is not the final amount: add the initial value when a final quantity is requested. Total distance or total variation instead integrates the absolute rate.

Formula

Formula and conditions

Q(b)-Q(a)=\int_a^b Q\prime(t)\,dt

Accumulating a rate gives signed change.

Use when: The rate is integrable on the interval.
Do not assume: Omitting the initial value confuses change with final amount.

A rate accumulates signed change

The rate 6−t stays positive on [0,4], so its integral adds 16 units. Rate times time has units of the accumulated quantity.

The interactive version of this figure is part of the full study guide.

A rate accumulates signed change: values

Case	Result
Series 1	$r(t)=6-t$

Use the structure

Identify the rate and units.
Set bounds.
Choose signed or absolute accumulation.
Add an initial value only for a final amount.

Keep these three ideas

Rate times width has quantity units.
Integration preserves sign.
Final equals initial plus net change.

Keep reading Unit 6: Integration and Accumulation of Change

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6.1 Exploring Accumulations of ChangeRead

6.2 Approximating Areas with Riemann Sums

6.3 Riemann Sums, Summation Notation, and Definite Integral Notation

6.4 The Fundamental Theorem of Calculus and Accumulation Functions

6.5 Interpreting the Behavior of Accumulation Functions Involving Area

6.6 Applying Properties of Definite Integrals

6.7 The Fundamental Theorem of Calculus and Definite Integrals

6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

6.9 Integrating Using Substitution

6.10 Integrating Functions Using Long Division and Completing the Square

6.11 Selecting Techniques for Antidifferentiation

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