# Integration and Accumulation of Change Overview

> AP Calculus AB · AP Calculus AB
> Source: https://www.owlsprep.com/study/ap-calculus-ab-u6-overview/
> Weight: 17-20% of total AP Calculus AB exam score

This unit introduces integration, the second core operation of calculus, connecting it to differentiation and the measurement of total accumulated change over an interval. Mastery of integration makes up a large portion of the AP exam.

**Prerequisites:** [Unit 2: Differentiation: Definition and Fundamental Properties](https://www.owlsprep.com/study/ap-calculus-ab-u2-overview/); [Unit 3: Differentiation: Composite, Implicit, and Inverse Functions](https://www.owlsprep.com/study/ap-calculus-ab-u3-overview/); [Limits and continuity for functions](https://www.owlsprep.com/study/ap-calculus-ab-u1-limits-continuity-overview/)

## Learning objectives

- Define integration as the inverse of differentiation and a tool to measure total accumulated change over an interval
- Approximate areas under curves and accumulated change using Riemann sums and numerical methods
- Apply both versions of the Fundamental Theorem of Calculus to relate derivatives and integrals
- Evaluate definite and indefinite integrals using basic rules and core integration techniques
- Analyze the graphical and numerical behavior of accumulation functions

## Unit at a Glance

We begin by connecting integration to the inverse of differentiation, introducing antiderivatives and basic indefinite integral rules. Next, we build intuition for integration as area under a curve, starting with approximations using Riemann sums before formalizing the definition of the definite integral.

The centerpiece of this unit is the Fundamental Theorem of Calculus (FTC), which unites differentiation and integration, and introduces accumulation functions that describe total change up to any point. We end by learning core integration techniques including u-substitution, and practice selecting the right approach for different integrands.

This unit progresses from basic definitions to applied techniques, with the following sub-topics:
- [AP Calculus AB Antiderivatives and indefinite integrals (basic rules)](https://www.owlsprep.com/study/ap-calculus-ab-u6-antiderivatives-and-indefinite-integrals/) — Learn what an antiderivative is and apply basic rules for computing indefinite integrals.
- [AP Calculus AB Approximating areas with Riemann sums](https://www.owlsprep.com/study/ap-calculus-ab-u6-approximating-areas-with-riemann-sums/) — Approximate the area under a curve using left, right, and midpoint Riemann sums.
- [AP Calculus AB Exploring accumulations of change](https://www.owlsprep.com/study/ap-calculus-ab-u6-exploring-accumulations-of-change/) — Connect definite integrals to real-world examples of total accumulated change over time.
- [AP Calculus AB FTC and definite integrals](https://www.owlsprep.com/study/ap-calculus-ab-u6-ftc-and-definite-integrals/) — Introduce the Fundamental Theorem of Calculus for evaluating definite integrals.
- [AP Calculus AB Fundamental Theorem of Calculus and accumulation functions](https://www.owlsprep.com/study/ap-calculus-ab-u6-fundamental-theorem-of-calculus-and/) — Explore accumulation functions defined by integrals and their derivatives.
- [AP Calculus AB Integration by substitution (u-sub)](https://www.owlsprep.com/study/ap-calculus-ab-u6-integration-by-substitution/) — Master u-substitution, the most common basic technique for integrating composite functions.
- [AP Calculus AB Integration with long division and completing the square](https://www.owlsprep.com/study/ap-calculus-ab-u6-integration-with-long-division-and/) — Learn to rewrite rational integrands using algebraic manipulation before integration.
- [AP Calculus AB Interpreting behavior of accumulation functions](https://www.owlsprep.com/study/ap-calculus-ab-u6-interpreting-behavior-of-accumulation-functions/) — Analyze the increasing/decreasing behavior and concavity of accumulation functions.
- [AP Calculus AB Properties of definite integrals](https://www.owlsprep.com/study/ap-calculus-ab-u6-properties-of-definite-integrals/) — Use properties of integrals to simplify calculations and combine integral results.
- [AP Calculus AB Riemann sums, summation notation, definite integral notation](https://www.owlsprep.com/study/ap-calculus-ab-u6-riemann-sums-summation-notation-definite/) — Formalize Riemann sums with summation notation and introduce definite integral notation.
- [AP Calculus AB Selecting techniques for antidifferentiation](https://www.owlsprep.com/study/ap-calculus-ab-u6-selecting-techniques-for-antidifferentiation/) — Practice choosing the correct integration strategy for a given integrand.

## Common pitfalls

- **Wrong:** Forgetting the constant of integration $+C$ when computing indefinite integrals
  - Why it fails: This common error leads to lost points on both multiple choice and free response AP exam questions
  - Correct: Always add $+C$ to the end of every indefinite integral solution.
- **Wrong:** Mixing up the two versions of the Fundamental Theorem of Calculus
  - Why it fails: Confusing the FTC for evaluating definite integrals with the FTC for differentiating accumulation functions leads to sign and derivative errors
  - Correct: Memorize the two forms separately and practice identifying which version applies to each problem.
- **Wrong:** Forgetting to substitute back for $x$ after u-substitution for indefinite integrals
  - Why it fails: Leaving the final answer in terms of $u$ results in an incorrect solution, even if intermediate steps are correct
  - Correct: Always reverse the u-substitution to write the final antiderivative in terms of the original variable.

## Cheatsheet

| Concept / Rule | Formula / Key Statement |
| --- | --- |
| Power Rule for Integration | $\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$ |
| Linearity of Integration | $ \\\text{af(x)} + bg(x)) dx = a \\f(x)dx + b \text{g(x)}dx + C$ |
| Left Riemann Sum | $L_n =  sum_{i=1}^n f(x_{i-1})\Delta x$ |
| Additivity of Integration | $ \ _a^c f(x)dx =  \ _a^b f(x)dx +  \ _b^c f(x)dx$ |
| Definite Integral Sign Property | $ \ _a^b f(x)dx = -  \ _b^a f(x)dx$ |
| FTC Part 1 (Evaluate Definite Integral) | $ \ _a^b f(x)dx = F(b) - F(a), \quad F'(x) = f(x)$ |
| FTC Part 2 (Differentiate Accumulation) | $ \frac{d}{dx}  \ _a^x f(t)dt = f(x)$ |
| Integration by Substitution Rule | $ \  f(g(x))g'(x)dx =  \  f(u)du$ |
| Total Change from Rate | $ \text{Total change from } a \text{ to } b =  \ _a^b \text{(rate of change)} dt$ |
| Antiderivative of $ \frac{1}{x}$ | $ \ \frac{1}{x} dx =  \ln\|x\| + C$ |

## What's next

Ready to start this unit? Begin with the first sub-topic on antiderivatives and indefinite integrals to build your foundational knowledge of integration. Once you complete all sub-topics in this unit, you will move on to Unit 7: Differential Equations, where you will apply integration to solve common differential equation problems.

- [AP Calculus AB Antiderivatives and indefinite integrals (basic rules)](https://www.owlsprep.com/study/ap-calculus-ab-u6-antiderivatives-and-indefinite-integrals/)
- [Exploring accumulations of change](https://www.owlsprep.com/study/ap-calculus-ab-u6-exploring-accumulations-of-change/)
- [Approximating Areas with Riemann Sums](https://www.owlsprep.com/study/ap-calculus-ab-u6-approximating-areas-with-riemann-sums/)

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