AP Calculus AB and BC

Unit 6: Integration and Accumulation of Change

Exam weighting: AB 15-20% · BC 15-20%

Unit 6 treats integration as accumulation: adding a rate over an interval gives total change. You learn Riemann sums, the Fundamental Theorem of Calculus linking integrals to antiderivatives, and how to choose an antidifferentiation technique. At 15-20% of both AB and BC, it is a top-weighted unit.

The single idea beneath all of Unit 6: a definite integral adds up a rate of change to give total change. If ff is a rate, then ∫abf(x) dx\int_a^b f(x)\,dx is the net accumulation from aa to bb, and geometrically it is the signed area between the graph and the xx-axis (Topic 6.1). Every other topic is a way to compute or interpret that one quantity.

The topics stack in three layers. First you approximate the area with Riemann sums (left, right, midpoint) and the trapezoidal sum, and decide whether each is an over- or underestimate (Topics 6.2 and 6.3). Then the Fundamental Theorem of Calculus turns that limit of sums into an exact value (Topics 6.4 through 6.7). Finally you build the toolkit of antiderivatives that makes the FTC usable (Topics 6.8 through 6.14).

The Fundamental Theorem of Calculus, both halves

Part 1 says differentiation undoes integration: for a continuous function ff, the accumulation function g(x)=∫axf(t) dtg(x)=\int_a^x f(t)\,dt has derivative g′(x)=f(x)g'(x)=f(x) (Topics 6.4 and 6.5). Part 2 says an antiderivative evaluates a definite integral: if ff is continuous and F′=fF'=f, then ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx=F(b)-F(a) (Topic 6.7). Part 1 tells you what an integral builds; Part 2 tells you how to compute one.

g′(x)=f(x)∫abf(x) dx=F(b)−F(a)g'(x) = f(x) \qquad \int_a^b f(x)\,dx = F(b)-F(a)

Method selection is the whole game in the back half of the unit, and Topic 6.14 exists only to practice it. Nobody hands you a labeled cue that says use substitution here, so the real skill is reading the integrand and matching it to a technique before you write anything down.

  • Basic rule or algebraic rewrite (Topics 6.8 and 6.10): the integrand is already a power, exponential, or trig form, or becomes one after long division or completing the square. Always try this first.
  • uu-substitution (Topic 6.9): the integrand holds an inner function together with a constant multiple of its derivative, like ∫2xcos⁡(x2) dx\int 2x\cos(x^2)\,dx. Let uu be the inner function, and for a definite integral change the limits to uu-values.
  • Integration by parts (Topic 6.11, BC only): a product of two unlike functions, such as xexx e^x or xln⁡xx\ln x, where one factor gets simpler when differentiated.
  • Partial fractions (Topic 6.12, BC only): a rational function whose denominator factors into distinct linear pieces.

On the exam Unit 6 carries 15-20% of the score on both AB and BC, tied for the heaviest weight. Multiple-choice questions pull Riemann sums from a table (Topic 6.2), differentiate an accumulation function (Topic 6.4), and evaluate definite integrals by the FTC (Topic 6.7). Free-response frequently gives a rate as a table or graph and asks for accumulated change with correct units (Topic 6.1). BC students also see integration by parts, partial fractions, and improper integrals (Topics 6.11 through 6.13).

Topics in this unit

Topic numbers and titles from the College Board Course and Exam Description.

  • 6.1Exploring Accumulations of Change
  • 6.2Approximating Areas with Riemann Sums
  • 6.3Riemann Sums, Summation Notation, and Definite Integral Notation
  • 6.4The Fundamental Theorem of Calculus and Accumulation Functions
  • 6.5Interpreting the Behavior of Accumulation Functions Involving Area
  • 6.6Applying Properties of Definite Integrals
  • 6.7The Fundamental Theorem of Calculus and Definite Integrals
  • 6.8Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
  • 6.9Integrating Using Substitution
  • 6.10Integrating Functions Using Long Division and Completing the Square
  • 6.11Integrating Using Integration by PartsBC only
  • 6.12Integrating Using Linear Partial FractionsBC only
  • 6.13Evaluating Improper IntegralsBC only
  • 6.14Selecting Techniques for Antidifferentiation

How to study this unit

  • Before integrating anything, run the Topic 6.14 checklist: can I use a basic rule, is there a $u$-substitution (an inner function and its derivative), or does it need integration by parts or partial fractions (BC)? Naming the method first prevents wasted work.
  • Lock down the over/underestimate logic for Riemann sums (Topic 6.2) by sketching: for an increasing function a left sum underestimates and a right sum overestimates, and a trapezoidal sum overestimates when the graph is concave up.
  • Drill the derivative-of-an-integral pattern from Topic 6.4 until it is automatic, including the chain rule when the upper limit is a function: $\frac{d}{dx}\int_a^{x^2} f(t)\,dt = f(x^2)\cdot 2x$.
  • On a definite-integral $u$-substitution (Topic 6.9), convert the limits to $u$-values instead of back-substituting to $x$; it is faster and kills a common sign error.
  • Attach units to every accumulation answer (Topic 6.1): a rate in liters per minute integrated over minutes gives liters, and FRQ graders dock points for missing or wrong units.

Guides for this unit

Tools and tables for this unit