# Integration and Accumulation of Change Overview

> AP Calculus BC · Unit 6: Integration and Accumulation of Change
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u6-overview/
> Weight: 17-20% of total AP exam score

This unit introduces integration as the inverse of differentiation and a tool for calculating net change and area. It builds core integration skills used across all remaining AP Calculus BC topics and makes up a large portion of the exam.

**Prerequisites:** [Unit 1: Limits and Continuity](https://www.owlsprep.com/study/ap-calculus-bc-u1-overview/); [Units 2-4: Differentiation: Rules and Applications](https://www.owlsprep.com/study/ap-calculus-bc-u4-overview/)

## Learning objectives

- Understand the inverse relationship between differentiation and integration
- Approximate areas and net change using Riemann sums and connect approximations to the definite integral
- Apply both parts of the Fundamental Theorem of Calculus to evaluate integrals and analyze accumulation functions
- Master core integration techniques for algebraic, trigonometric, and transcendental functions
- Evaluate improper integrals and interpret their convergence (BC only)

## Unit at a Glance

This unit follows a logical learning arc that connects two core ideas of integration: integration as the inverse of differentiation, and integration as a method to calculate accumulated change over an interval. We start with basic antiderivatives and area approximation, then formalize the definition of the definite integral before introducing the Fundamental Theorem of Calculus (FTC) that unites these two core ideas.

After building a solid conceptual foundation, we learn a range of integration techniques for different types of functions, ending with BC-exclusive topics like improper integrals, integration by parts, and partial fractions. AP exam questions frequently draw on connections between multiple topics in this unit, so prioritizing conceptual understanding alongside technical skill is critical for success.

Below are the sub-topics you will explore in this unit:
- [AP Calculus BC Antiderivatives and indefinite integrals (basic rules)](https://www.owlsprep.com/study/ap-calculus-bc-u6-antiderivatives-and-indefinite-integrals/) — Introduces antiderivatives as the inverse of derivatives and basic rules for indefinite integrals.
- [AP Calculus BC Approximating areas with Riemann sums](https://www.owlsprep.com/study/ap-calculus-bc-u6-approximating-areas-with-riemann-sums/) — Teaches area approximation using left, right, midpoint, and trapezoidal Riemann sums.
- [AP Calculus BC Exploring accumulations of change](https://www.owlsprep.com/study/ap-calculus-bc-u6-exploring-accumulations-of-change/) — Connects integration to the concept of net accumulation of change over an interval.
- [AP Calculus BC FTC and definite integrals](https://www.owlsprep.com/study/ap-calculus-bc-u6-ftc-and-definite-integrals/) — Covers the second part of the FTC for evaluating definite integrals from antiderivatives.
- [AP Calculus BC Fundamental Theorem of Calculus and accumulation functions](https://www.owlsprep.com/study/ap-calculus-bc-u6-fundamental-theorem-of-calculus-and/) — Explores the first part of the FTC and how it defines derivatives of accumulation functions.
- [AP Calculus BC Improper integrals (BC only)](https://www.owlsprep.com/study/ap-calculus-bc-u6-improper-integrals/) — BC-only topic covering integrals with infinite bounds or discontinuities and convergence testing.
- [AP Calculus BC Integration by parts (BC only)](https://www.owlsprep.com/study/ap-calculus-bc-u6-integration-by-parts/) — BC-only integration technique for products of functions, derived from the product rule.
- [AP Calculus BC Integration by substitution (u-sub)](https://www.owlsprep.com/study/ap-calculus-bc-u6-integration-by-substitution/) — Core integration technique for composite functions, the most widely used basic integration method.
- [AP Calculus BC Integration using partial fractions (BC only)](https://www.owlsprep.com/study/ap-calculus-bc-u6-integration-using-partial-fractions/) — BC-only technique for integrating rational functions by breaking them into simpler terms.
- [AP Calculus BC Integration with long division and completing the square](https://www.owlsprep.com/study/ap-calculus-bc-u6-integration-with-long-division-and/) — Covers algebraic manipulation of improper rational functions to prepare for integration.
- [AP Calculus BC Interpreting behavior of accumulation functions](https://www.owlsprep.com/study/ap-calculus-bc-u6-interpreting-behavior-of-accumulation-functions/) — Analyzes the increasing/decreasing behavior and concavity of accumulation functions.
- [AP Calculus BC Properties of definite integrals](https://www.owlsprep.com/study/ap-calculus-bc-u6-properties-of-definite-integrals/) — Covers core properties including symmetry, interval splitting, and constant multiple rules.
- [AP Calculus BC Riemann sums, summation notation, definite integral notation](https://www.owlsprep.com/study/ap-calculus-bc-u6-riemann-sums-summation-notation-definite/) — Connects the limit of Riemann sums to the formal definition of the definite integral.
- [AP Calculus BC Selecting techniques for antidifferentiation](https://www.owlsprep.com/study/ap-calculus-bc-u6-selecting-techniques-for-antidifferentiation/) — Teaches how to choose the correct integration method based on the integrand form.

## Common pitfalls

- **Wrong:** Applying the Fundamental Theorem of Calculus to discontinuous or unbounded integrands
  - Why it fails: The standard FTC only works for continuous functions over closed bounded intervals, so this leads to incorrect results
  - Correct: Always check for discontinuities or infinite bounds, and use improper integral methods when needed
- **Wrong:** Failing to adjust the bounds of a definite integral after u-substitution
  - Why it fails: Leaving original bounds in terms of x after substituting u leads to wrong numerical values
  - Correct: Always convert bounds to the new variable u when working with definite integrals
- **Wrong:** Forgetting the constant of integration +C for indefinite integrals
  - Why it fails: Indefinite integrals represent a family of antiderivatives, not a single function
  - Correct: Always add +C to the result of every indefinite integral calculation

## Cheatsheet

| Key Concept | Formula/Rule |
| --- | --- |
| Indefinite Power Rule | $\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$ |
| General Riemann Sum | $\text{Area} \approx \sum_{i=1}^n f(x_i^*) \Delta x$ |
| FTC Part 1 (Derivative of Accumulation) | $\frac{d}{dx} \int_a^x f(t) dt = f(x)$ |
| FTC Part 2 (Evaluate Definite Integral) | $\int_a^b f(x) dx = F(b) - F(a), \quad F'(x) = f(x)$ |
| U-Substitution Rule | $\int f(g(x))g'(x) dx = \int f(u) du$ |
| Integration by Parts | $\int u \, dv = uv - \int v \, du$ |
| Improper Integral (Infinite Bound) | $\int_a^\infty f(x) dx = \lim_{b \to \infty} \int_a^b f(x) dx$ |
| Partial Fraction Decomposition (Linear Factors) | $\frac{px+q}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$ |

## What's next

Start your study of this unit with the first sub-topic on antiderivatives and indefinite integrals to build your foundational understanding of integration. Once you complete all sub-topics in this unit, you will move on to Unit 7: Differential Equations, where you will apply your new integration skills to solve separable differential equations and model dynamic systems.

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

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