AP Calculus AB/BC Unit 6 Cheat Sheet: Integration & Accumulation of Change

The basics

What it covers: Riemann sums, the definite integral, the Fundamental Theorem of Calculus, accumulation functions, antiderivatives, u-substitution, and (BC) parts, partial fractions, and improper integrals.

Exam weight: About 17–20% of both the AB and BC exams — one of the largest units.

The big question: How do we measure accumulated change, and how do integration and differentiation connect?

Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.

Key topics at a glance

Accumulation of Change

The area under a rate curve is total change: total change = ∫₀ᵇ rate dt. A definite integral is signed area.

Riemann Sums

Approximate area with rectangles Σ f(xᵢ)Δx. Left under-, right over-estimates for increasing f; midpoint and trapezoid are usually better.

Definite Integral

∫₀ᵇ f(x) dx = lim Σ f(xᵢ)Δx as n → ∞ — the exact signed area.

FTC (Evaluation)

∫₀ᵇ f(x) dx = F(b) − F(a), where F′ = f. Integrate, then evaluate at the bounds.

FTC (Accumulation)

If g(x) = ∫₀ˣ f(t) dt, then g′(x) = f(x). With upper limit u(x): g′ = f(u(x))·u′(x).

Antiderivative Rules

∫xⁿ dx = xⁿ⁺¹/(n+1)+C; ∫(1/x)dx = ln|x|+C; ∫eᵇ dx = eᵇ+C; ∫cos = sin+C; ∫sin = −cos+C.

u-Substitution

Reverse the chain rule: let u = inner, du = u′ dx, rewrite, integrate, back-substitute. Convert the limits for definite integrals.

Integral Properties

Reverse limits negate; adjacent intervals add; constants factor out; ∫₀₀ f = 0.

The key terms and rules you must know

Definite integral — signed area / accumulated change over [a,b].
Riemann sum — Σ f(xᵢ)Δx approximating area.
Left / right / midpoint / trapezoid — the approximation methods.
Antiderivative — F with F′ = f; ∫f dx = F + C.
FTC (evaluation) — ∫₀ᵇ f = F(b) − F(a).
FTC (accumulation) — d/dx ∫₀ˣ f(t)dt = f(x).
Power rule — ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1).
∫(1/x)dx — ln|x| + C.
u-substitution — reverse chain rule; u = inner function.
Integration by parts (BC) — ∫u dv = uv − ∫v du.

Key themes to remember

Integration measures accumulation. The integral of a rate is total change.
Riemann sums → the integral. Rectangles in the limit become exact area.
The FTC links the two calculi. Derivatives and integrals undo each other.
Antidifferentiation is a toolkit. Basic rules, then u-sub, then BC techniques.
Signed area counts sign. Area below the x-axis contributes negatively.

Common exam traps

Don’t forget + C on indefinite integrals.
Change the limits (or back-substitute) with u-substitution in definite integrals.
Left/right over/under depends on increasing vs. decreasing. Sketch to be sure.
∫(1/x)dx = ln|x| + C, not x⁰/0 — the power rule fails at n = −1.
Signed area ≠ total area. For total area, integrate |f| or split at the zeros.
FTC with a variable upper limit uses the chain rule: multiply by u′(x).
Match du exactly — account for constant factors when substituting.