Unit Overview
Integration and Accumulation of Change Overview
AP Calculus ABΒ· 5 min read π 17-20% of total AP Calculus AB exam score
1. 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)
Learn what an antiderivative is and apply basic rules for computing indefinite integrals.
β β β± 10 min
AP Calculus AB Approximating areas with Riemann sums
Approximate the area under a curve using left, right, and midpoint Riemann sums.
β β β± 8 min
AP Calculus AB Exploring accumulations of change
Connect definite integrals to real-world examples of total accumulated change over time.
β β β β± 7 min
AP Calculus AB FTC and definite integrals
Introduce the Fundamental Theorem of Calculus for evaluating definite integrals.
β β β β± 9 min
AP Calculus AB Fundamental Theorem of Calculus and accumulation functions
Explore accumulation functions defined by integrals and their derivatives.
β β β β β± 10 min
AP Calculus AB Integration by substitution (u-sub)
Master u-substitution, the most common basic technique for integrating composite functions.
β β β β β± 12 min
AP Calculus AB Integration with long division and completing the square
Learn to rewrite rational integrands using algebraic manipulation before integration.
β β β β β± 8 min
AP Calculus AB Interpreting behavior of accumulation functions
Analyze the increasing/decreasing behavior and concavity of accumulation functions.
β β β β β± 9 min
AP Calculus AB Properties of definite integrals
Use properties of integrals to simplify calculations and combine integral results.
β β β± 7 min
AP Calculus AB Riemann sums, summation notation, definite integral notation
Formalize Riemann sums with summation notation and introduce definite integral notation.
β β β β± 10 min
AP Calculus AB Selecting techniques for antidifferentiation
Practice choosing the correct integration strategy for a given integrand.
β β β β β± 8 min
2. Common Pitfalls
Wrong move:
Forgetting the constant of integration when computing indefinite integrals
Why:
This common error leads to lost points on both multiple choice and free response AP exam questions
Correct move:
Always add to the end of every indefinite integral solution.
Wrong move:
Mixing up the two versions of the Fundamental Theorem of Calculus
Why:
Confusing the FTC for evaluating definite integrals with the FTC for differentiating accumulation functions leads to sign and derivative errors
Correct move:
Memorize the two forms separately and practice identifying which version applies to each problem.
Wrong move:
Forgetting to substitute back for after u-substitution for indefinite integrals
Why:
Leaving the final answer in terms of results in an incorrect solution, even if intermediate steps are correct
Correct move:
Always reverse the u-substitution to write the final antiderivative in terms of the original variable.
3. Quick Reference Cheatsheet
Concept / Rule | Formula / Key Statement |
|---|---|
Power Rule for Integration | |
Linearity of Integration | $ \ ext{af(x)} + bg(x)) dx = a \ (x)dx + b \ ext{g(x)}dx + C$ |
Left Riemann Sum | $L_n = sum_{i=1}^n f(x_{i-1})\Delta x$ |
Additivity of Integration | $ |
Definite Integral Sign Property | $ |
FTC Part 1 (Evaluate Definite Integral) | $ |
FTC Part 2 (Differentiate Accumulation) | $ \frac{d}{dx} |
Integration by Substitution Rule | $ |
Total Change from Rate | $ \text{Total change from } a \text{ to } b = |
Antiderivative of $ \frac{1}{x}$ | $ |
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.