AP Calc AB · Unit 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.

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Why this unit 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.

Understand the core ideas

Integration in AP Calculus AB begins as accumulation. A Riemann sum adds many small products of rate times width, which approximates total change or signed area. Refining the partition improves the approximation, and in the limit you obtain the definite integral. Unlike raw geometric area, a definite integral can be negative when the graph lies below the x-axis, so it represents net accumulation. This sign convention is essential in motion and economics contexts.

Antiderivatives reverse differentiation. If F'(x) = f(x), then integral f(x) dx = F(x) + C for indefinite forms. The Fundamental Theorem of Calculus links both worlds: for accumulation function A(x) = ∫axf(t) dt\int_{a}^{x} f(t) \, dt, we have A'(x) = f(x). Also, ∫abf(x) dx\int_{a}^{b} f(x) \, dx = F(b) - F(a) when F is an antiderivative. Substitution in AB is reverse chain rule, useful when an inner expression has a derivative factor present. Keep track of bounds carefully if you switch to definite integral form with u bounds.

Key terms

Riemann sum
A finite sum that approximates a definite integral by adding function values times subinterval widths.
Antiderivative
A function F whose derivative is a given function f, so F'(x) = f(x).
Definite integral
A number representing net accumulation of a function over an interval.
Fundamental Theorem of Calculus
The theorem connecting derivatives and integrals through accumulation functions and endpoint evaluation.

Use both parts of the Fundamental Theorem

Let A(x) = ∫1x(t2+2t) dt\int_{1}^{x} (t^2 + 2t) \, dt. Find A'(x) and A(3).

  1. By FTC Part 1, A'(x) equals the integrand at x, so A′(x)=x2+2xA'(x) = x^2 + 2x.
  2. Find an antiderivative of t2+2tt^2 + 2t: F(t)=t3/3+t2F(t) = t^3/3 + t^2.
  3. Apply FTC Part 2: A(3) = F(3) - F(1).
  4. Compute: F(3) = 9 + 9 = 18, F(1) = 1/3 + 1 = 4/3, so A(3) = 18 - 4/3 = 50/3.
Result: A′(x)=x2+2xA'(x) = x^2 + 2x and A(3) = 50/3.

A common misconception

Claim: A definite integral always gives physical area, so it cannot be negative.

Correction: A definite integral gives net signed accumulation. Values below the axis contribute negatively.

Lessons in this unit

  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.

Unit checkpoint

Evaluate ∫02(3x2+1) dx\int_{0}^{2} (3x^2 + 1) \, dx.

An antiderivative is x3+xx^3 + x. Evaluate: (8 + 2) - (0 + 0) = 10.

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