AP Calculus BC · Unit 6

Integration and Accumulation of Change — practice questions organized by learning target

946 questions · source: apfive.com

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All the key formulas for Unit 6 in one place.

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Learning Targets (College Board Unit 6)

1.Exploring Accumulations of Change — Approximate and interpret accumulated change from a rate of change.
2.Approximating Areas with Riemann Sums — Approximate area under a curve using left, right, and midpoint Riemann sums.
3.Riemann Sums, Summation Notation, and Definite Integral Notation — Connect Riemann sums, sigma notation, and the integral sign to the definite integral.
4.The Fundamental Theorem of Calculus and Accumulation Functions — Relate accumulation functions to their derivatives using the FTC.
5.Interpreting the Behavior of Accumulation Functions Involving Area — Analyze functions defined by integrals: increasing/decreasing, concavity, and extrema.
6.Applying Properties of Definite Integrals — Use additivity, reversing limits, and even/odd symmetry to compute integrals.
7.The Fundamental Theorem of Calculus and Definite Integrals — Evaluate definite integrals with antiderivatives: F(b) - F(a).
8.Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation — Find antiderivatives and indefinite integrals using basic rules and the +C.
9.Integrating Using Substitution — Evaluate indefinite and definite integrals with u-substitution.
10.Integrating Functions Using Long Division and Completing the Square — Rewrite rational expressions with long division or completing the square before integrating.
11.Using Integration by Parts (BC only) — Integrate products of functions using ∫u dv = uv - ∫v du.
12.Integrating Using Partial Fractions (BC only) — Integrate rational functions by decomposing them into partial fractions.
13.Evaluating Improper Integrals (BC only) — Evaluate integrals with infinite limits or infinite discontinuities.
14.Selecting Techniques for Antidifferentiation — Choose the appropriate integration technique for a given integral.

Practice by Topic

Power Rule & Polynomial Integrals

The power rule, polynomial/linear/power functions, and basic algebraic integration.

60 questions Easy

Left Riemann Sums

Computing left Riemann sums from tables, formulas, and graphs.

21 questions Easy

Evaluating Definite Integrals

Computing the value of a definite integral using an antiderivative.

30 questions Easy

FTC Part 1: Evaluate Definite Integrals

Using the first part of the FTC (antiderivatives) to evaluate definite integrals.

34 questions Easy

Definite Integral Notation & Definition

Reading integral notation, the limit definition, and geometric meaning of a definite integral.

27 questions Easy

Net Change & Motion

Using integrals for net change: position, velocity, displacement, distance, and rates.

33 questions Easy

Applications: Area, Volume & More

Area between curves, volumes of revolution, average value, and arc length.

12 questions Easy

Right Riemann Sums

Computing right Riemann sums from tables, formulas, and graphs.

23 questions Easy

Trapezoidal Rule

Trapezoidal sums and the trapezoidal rule, including from tabular data.

31 questions Easy

Exponential & Logarithmic Integrals

Integrating exponential, logarithmic, and reciprocal (1/x) functions.

22 questions Easy

Riemann Sum Definition & Limits

The limit definition of the definite integral and expressing limits as Riemann sums.

38 questions Easy

U-Substitution

Choosing u, changing limits of integration, and substitution technique (incl. trig substitution).

87 questions Easy

Antiderivatives & Integration Concepts

Definitions, the constant of integration, and choosing an integration technique.

47 questions Easy

Accumulation Functions

Functions defined by integrals, their properties, behavior, and extrema.

41 questions Medium

Trigonometric Integrals

Integrating trigonometric and inverse-trigonometric functions.

23 questions Medium

Properties of Definite Integrals

Additivity, reversing limits, even/odd symmetry, piecewise functions, and other properties.

54 questions Medium

Midpoint Riemann Sums

Computing midpoint Riemann sums and midpoint-rule approximations.

24 questions Medium

Approximation Accuracy & Concepts

Over/underestimation, comparing approximation methods, error, and Simpson's rule.

36 questions Medium

Initial Value Problems

Solving for functions given an antiderivative and initial condition (motion included).

28 questions Hard

Solving Integral Equations

Solving for unknown constants, limits of integration, or the function itself.

29 questions Hard

Improper Integrals

Infinite limits of integration, infinite discontinuities, and convergence/divergence.

53 questions Hard

FTC Part 2 & Chain Rule

The second part of the FTC: differentiating integrals, variable limits, and the chain rule.

40 questions Hard

Partial Fractions & Rational Functions

Partial fraction decomposition, polynomial long division, and rational-function techniques.

87 questions Hard

Integration by Parts

The parts formula, LIATE strategy, tabular method, and repeated integration by parts.

66 questions Hard