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AP Calc AB · Chapter 6 of 8

Integration and Accumulation of Change

Introduce antiderivatives, Riemann sums, and definite integrals, then use the Fundamental Theorem of Calculus to connect accumulation with derivatives.

Why this chapter matters

Integration formalizes accumulated change, so you can recover total amounts from rate information.

What you will learn

  • Approximate area and accumulation with left, right, and midpoint Riemann sums.
  • Compute antiderivatives and evaluate definite integrals.
  • Apply both parts of the Fundamental Theorem of Calculus.

Lessons in this chapter

  1. Area accumulation and Riemann sumsInterpret sigma-style sums as approximations to accumulated quantity.
  2. Antiderivatives and indefinite integralsUse reverse differentiation patterns to find families of functions.
  3. Definite integrals and net changeEvaluate integrals as signed accumulation over an interval. Read the full guide →
  4. Substitution as reverse chain ruleTransform integrals into simpler variables and bounds. Read the full guide →

Study task

Given a velocity function v(t), estimate displacement on [0, 4] with midpoint sums, then compare to the exact definite integral.

Chapter checkpoint

Evaluate integral from 0 to 2 of (3x^2 + 1) dx.

An antiderivative is x^3 + x. Evaluate: (8 + 2) - (0 + 0) = 10.