Unit 6: Integration & Accumulation of Change
Unleash the Power of Integrals: Master Accumulation, Area, and Advanced AP® Calculus Strategies
Travel from Riemann Sums to the Fundamental Theorem—across antiderivatives, substitution, and real-world problem-solving. Everything you need from setup to advanced BC integration skills, with interactive lessons and proven strategies!
📚 Unit Overview
Unit 6: Integration & Accumulation of Change unlocks the other half of calculus, introducing you to accumulation, area, total change, and the definitive power of antidifferentiation. You'll master definite and indefinite integrals, numeric and analytic approaches, plus all major AP® integral techniques.
Through 14 sequential topics, you'll learn both AB and BC level content, from foundational principles to advanced integration methods—every topic is filled with formula sheets, memory tips, and AP-aligned examples.
🎯 Key Concepts You'll Master
- Accumulation & Area: Key ideas of net change, total area under curves, and integrals as summing processes
- Riemann Sums: Left, right, midpoint, and trapezoidal approximations using summations
- Integral Notation: Proper setup, properties, and evaluation with definite/indefinite integrals
- The Fundamental Theorem of Calculus: Linking derivatives, integrals, and accumulation functions
- Properties of Integrals: Linearity, additivity, and symmetry for easy computation
- Antiderivatives: Building expressions for reversing differentiation
- Substitution: Efficient integration using the reverse Chain Rule
- Long Division & Completing the Square: Special algebraic techniques for difficult integrals
- Integration by Parts, Partial Fractions, and Improper Integrals (BC): Powerful advanced BC tools
- Selecting Techniques: How to choose the right method for different function types
- AP® Exam Tactics: Setup, precise notation, error avoidance, and maximizing integral points
🎓 Learning Objectives
After Unit 6, you will be able to:
- Interpret and set up accumulation problems and area under curve scenarios
- Approximate integrals numerically using Riemann sums
- Correctly use and evaluate both definite and indefinite integral notation
- Apply the Fundamental Theorem of Calculus, both parts I and II
- Calculate integrals using substitution and antiderivative rules
- Choose and carry out advanced BC techniques: long division, completing the square, parts, partial fractions, improper integrals
- Select the best antidifferentiation method for a given function or problem
- Avoid classic AP® errors and maximize exam points with proven strategies
📖 Complete Topic Guide (14 Lessons)
Click any topic below to access detailed formula sheets, worked examples, AP® strategies, and review exercises:
Exploring Accumulations of Change
Understand how calculus measures total change and area using integration.
Explore Topic 6.1 →Approximating Areas with Riemann Sums
Learn and apply left, right, midpoint, and trapezoidal sum approximations and their interpretations.
Explore Topic 6.2 →Riemann Sums, Summation Notation, and Definite Integral Notation
Transition from discrete sums to continuous notation and meaningful computation with the definite integral.
Explore Topic 6.3 →The Fundamental Theorem of Calculus and Accumulation Functions
See how differentiation and integration are perfectly linked by this central result.
Explore Topic 6.4 →Interpreting the Behavior of Accumulation Functions Involving Area
Analyze area and accumulation scenarios, and interpret real-world calculus meaning behind areas.
Explore Topic 6.5 →Applying Properties of Definite Integrals
Use linearity, additivity, and properties for fast and accurate integral evaluation.
Explore Topic 6.6 →The Fundamental Theorem of Calculus and Definite Integrals
Apply FTC to evaluate definite integrals step-by-step, including graphical and symbolic approaches.
Explore Topic 6.7 →Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
Develop your basic antiderivatives and learn indefinite integration notation and skills.
Explore Topic 6.8 →Integrating Using Substitution
Master the reverse chain rule to efficiently compute tricky integrals.
Explore Topic 6.9 →Integrating Functions Using Long Division and Completing the Square
Use advanced algebraic techniques to make difficult integrals tractable.
Explore Topic 6.10 →Integrating Using Integration by Parts (BC only)
Achieve greater integration power with the classic product rule reversal strategy.
Explore Topic 6.11 →Using Linear Partial Fractions (BC only)
Decompose and integrate rational functions for highly effective AP® BC solutions.
Explore Topic 6.12 →Evaluating Improper Integrals (BC only)
Take on infinite intervals and discontinuities, learning the logic and AP significance of improper integrals.
Explore Topic 6.13 →Selecting Techniques for Antidifferentiation
Learn how to identify and select the best antiderivative strategy for the integral at hand.
Explore Topic 6.14 →🌟 Why Unit 6 Matters
Unit 6 is calculus power in full: It brings together area, accumulation, total change, and a strategic arsenal of integration tools—core for both AP® Calculus and real STEM applications in science, engineering, and beyond.
- Connecting change: Integrals relate to velocity, distance, growth, and total phenomena in life and science
- Heavy AP® coverage: About 17-20% of exam points depend on integration, computation, and accumulation skills
- Mathematical reasoning: Integrating theorems, examples, and algebra deepen understanding across all units
- Advanced skillset: Techniques here prepare you for university-level calculus and multivariable math
✏️ AP® Exam Success: Unit 6 Strategy
How Unit 6 Appears on the AP® Calculus Exam:
Multiple Choice Questions (MCQ):
- Riemann sum approximations and interpreting definite integrals as area
- Applying the Fundamental Theorem directly and graphically
- Integral properties and short-cut evaluation
- Choosing and implementing proper technique (substitution, parts, etc.)
- Recognizing antiderivatives in context
- Advanced: partial fractions, improper integrals, and strategic method selection (BC)
Free Response Questions (FRQ):
- Complete area/accumulation problems with setup and AP® justification
- Detailed FTC applications in full symbolic and numeric contexts
- Complex integration steps and combination of techniques (especially BC)
- Prediction and interpretation of area, total, and net change
Key Success Strategies:
- Memorize all basic antiderivative forms & properties for polynomials, exponentials, trig, and rational functions
- Show clear substitution setup, u-variable, and endpoints
- Practice all computational techniques—not just know them, but recognize when to apply
- Use endpoint patterns & AP® notation precisely
- Avoid AP® errors: Watch sign and constant errors, pay attention to domain issues in improper integrals
📅 Recommended Study Path
Best progression for Unit 6 mastery:
- Week 1: Accumulation & Riemann Sums (Topics 6.1-6.3)
- Foundations of area, summation, and integral notation
- Week 2: FTC & Area Analysis (Topics 6.4-6.7)
- Apply the Fundamental Theorem in all problem forms
- Interpreting areas and integral properties in calculator and real contexts
- Week 3: Antiderivatives & Substitution (Topics 6.8-6.9)
- Learn and practice all basic indefinite skills and substitution approaches
- Week 4: Advanced (Topics 6.10-6.14, BC Only)
- Use advanced algebraic and AP® BC integration skills
- Practice selection and strategic application of all key methods
- Review mixed methods and integral problem types before the exam
🎁 What's Included in Each Topic Page
Every topic page includes:
- ✅ Formula & Integration Tables: Every major method and shortcut at your fingertips
- ✅ Worked Examples: Step-by-step AP® solutions and variety drills
- ✅ Memory Cues & Pitfalls: Avoid common errors and lock in key forms
- ✅ Exam Strategies: Integral error-checkers and scoring maximizers
- ✅ Practice Sets: For every method, from setup to advanced
- ✅ Summary Boxes: Visual quick reference and AP alert points
- ✅ Graph-Rich Layout: Area and accumulation visuals throughout
- ✅ SEO Focused: Keyword-packed explanations for fast exam and search prep
🚀 Unlock Integral Power Now
Make integrals second nature—build your calculus strength across accumulation, area, and every AP® technique in Unit 6.
Click any topic above to dive in! Each lesson is structured for AP® clarity, exam speed, and actual mastery. Unit 6 is the bridge to true calculus comprehension.