---
title: "AP Calc AB/BC Unit 6 Review: Integration & Accumulation"
description: "AP Calculus AB/BC Unit 6 covers Evaluating Improper Integrals and Integrating Using Substitution. Study guides, practice questions, and key terms."
canonical: "https://fiveable.me/ap-calc/unit-6"
type: "unit"
subject: "AP Calculus AB/BC"
unit: "Unit 6 – Integration and Accumulation of Change"
---

# AP Calc AB/BC Unit 6 Review: Integration & Accumulation

## Overview

Unit 6 introduces integration as the accumulation of change, establishes the Fundamental Theorem of Calculus, and develops a full toolkit of antidifferentiation techniques including u-substitution, long division, completing the square, and (BC only) integration by parts, partial fractions, and improper integrals.

## AP CED Alignment

This unit hub is organized around AP Course and Exam Description topics, skills, and exam task types when they are available in the source data.
- 6.1: Exploring Accumulations of Change
- 6.2: Approximating Areas with Riemann Sums
- 6.3: Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4: The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5: Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6: Applying Properties of Definite Integrals
- 6.7: The Fundamental Theorem of Calculus and Definite Integrals
- 6.8: Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9: Integrating Using Substitution
- 6.10: Integrating Functions Using Long Division and Completing the Square
- 6.11: Integrating Using Integration by Parts (BC Only)
- 6.12: Integrating Using Linear Partial Fractions (BC Only)
- 6.13: Evaluating Improper Integrals (BC Only)
- 6.14: Selecting Techniques for Antidifferentiation
- 6.1: Accumulation of Change and Area
- 6.2-6.3: Riemann Sums and Definite Integral Notation
- 6.4-6.5: Accumulation Functions and FTC Part 1
- 6.6-6.7: Properties of Definite Integrals and FTC Part 2
- 6.8-6.9: Basic Antiderivative Rules and u-Substitution
- 6.10: Long Division and Completing the Square
- 6.11: Integration by Parts (BC Only)
- 6.12: Linear Partial Fractions (BC Only)
- 6.13: Improper Integrals (BC Only)
- 6.14: Selecting Antidifferentiation Techniques
- Practice 2 - Connecting Representations
- Practice 3 - Justification
- Practice 1 - Implementing Mathematical Processes
- FRQs – No graphing calculator
- FRQs – Graphing calculator required

## Topics

- [6.1: Exploring Accumulations of Change](/ap-calc/unit-6/integration-accumulation-change/study-guide/NWRV9MaRJO4Eno32l5Xp): Area between a rate-of-change graph and the x-axis equals accumulated change. Positive rate means positive accumulation; negative rate means negative accumulation. Simple shapes can be evaluated with geometry. Units equal rate unit times input unit.
- [6.2: Approximating Areas with Riemann Sums](/ap-calc/unit-6/approximating-areas-with-riemann-sums/study-guide/juN9YbvFYlJtpsMl): Left, right, midpoint, and trapezoidal sums approximate a definite integral using rectangles or trapezoids. Over- and underestimate behavior depends on whether the function is increasing, decreasing, concave up, or concave down. Works with graphs, tables, equations, or verbal descriptions.
- [6.3: Riemann Sums, Summation Notation, and Definite Integral Notation](/ap-calc/unit-6/riemann-sums-summation-notation-definite-integral-notation/study-guide/RnM03H2k6l3ewvxX): A Riemann sum in sigma notation sums f(x_i*) times delta x_i over all subintervals. As the maximum subinterval width approaches zero, this limit equals the definite integral from a to b of f(x) dx. You should be able to translate between the limit of a sum and integral notation.
- [6.4: The Fundamental Theorem of Calculus and Accumulation Functions](/ap-calc/unit-6/fundamental-theorem-calculus-accumulation-functions/study-guide/TyDEzN9M5wieA4Kw): FTC Part 1: d/dx of the integral from a to x of f(t) dt = f(x). Accumulation functions g(x) = integral from a to x of f(t) dt are differentiable, with g(a) = 0. When the upper limit is h(x), apply the chain rule to get f(h(x)) times h'(x).
- [6.5: Interpreting the Behavior of Accumulation Functions Involving Area](/ap-calc/unit-6/interpreting-behavior-accumulation-functions-involving-area/study-guide/uVNoabC47nUgOdjr): Read g(x) = integral from a to x of f(t) dt from the graph of f. g increases where f > 0, decreases where f < 0, has extrema where f changes sign, and changes concavity where f changes from increasing to decreasing.
- [6.6: Applying Properties of Definite Integrals](/ap-calc/unit-6/applying-properties-definite-integrals/study-guide/lUbcVbDG5QVysAn9): Key properties: reversing limits changes sign, additivity over adjacent intervals, constant multiples and sums distribute. Geometric evaluation works when the graph forms triangles, rectangles, or semicircles. Integrals extend to functions with removable or jump discontinuities.
- [6.7: The Fundamental Theorem of Calculus and Definite Integrals](/ap-calc/unit-6/fundamental-theorem-calculus-definite-integrals/study-guide/fEGd7E9gbOH8EtCf): FTC Part 2: the integral from a to b of f(x) dx = F(b) minus F(a) for any antiderivative F of f. Requires f to be continuous on [a, b]. The constant of integration cancels in the subtraction.
- [6.8: Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation](/ap-calc/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3): Indefinite integral notation: integral of f(x) dx = F(x) + C. Core antiderivatives include the reverse power rule, e^x, 1/x, sin x, cos x, sec^2 x, 1/(1+x^2), and 1/sqrt(1-x^2). Differentiation rules run in reverse.
- [6.9: Integrating Using Substitution](/ap-calc/unit-6/integrating-using-substitution/study-guide/Oa1yaKFA8AzaadQq): U-substitution reverses the chain rule. Set u = inner function, replace g'(x) dx with du, integrate in u, back-substitute. For definite integrals, convert the limits to u-values and skip back-substitution.
- [6.10: Integrating Functions Using Long Division and Completing the Square](/ap-calc/unit-6/integrating-functions-using-long-division-completing-square/study-guide/ju79RFY6f5aKWjFK): Use polynomial long division when the numerator degree is greater than or equal to the denominator degree. Complete the square on irreducible quadratic denominators to reach arctan or arcsin antiderivative forms.
- [6.11: Integrating Using Integration by Parts (BC Only)](/ap-calc/unit-6/integrating-using-integration-by-parts/study-guide/O4P3LchNoZnWElf8zETV): Formula: integral of u dv = uv minus the integral of v du. Use LIATE to choose u. Repeat for higher-degree polynomial factors; use the cyclic method when integration by parts returns the original integral.
- [6.12: Integrating Using Linear Partial Fractions (BC Only)](/ap-calc/unit-6/using-linear-partial-fractions/study-guide/VNjHMatlmaZoFyt7yLPH): Decompose a proper rational function with distinct linear factors into A/(ax + b) terms. Solve for constants by substituting roots. Each term integrates to a natural log. Apply long division first if the function is improper.
- [6.13: Evaluating Improper Integrals (BC Only)](/ap-calc/unit-6/evaluation-improper-integrals/study-guide/DRGur03jVQG3WXLiNyuz): Replace an infinite bound or a point of unboundedness with a limit variable. Integrate normally, then evaluate the limit. Finite limit means convergence; infinite or nonexistent limit means divergence. Split at interior asymptotes.
- [6.14: Selecting Techniques for Antidifferentiation](/ap-calc/unit-6/selecting-techniques-for-antidifferentiation/study-guide/oTEgMFOzDwdVR7OgZzk1): Identify integrand structure before computing. Match the structure to the correct technique: basic rules, u-substitution, long division, completing the square, or (BC) integration by parts, partial fractions, or improper integral limits.

## Hardest Topics And Analytics

Snapshot: practice snapshot
This snapshot uses Fiveable practice activity to show where students tend to miss questions and which review moves are worth prioritizing first.
- **55% average MCQ accuracy** (Across 7.5k multiple-choice practice attempts for this unit.)
- **7.5k MCQ attempts** (Practice activity included in this snapshot.)
- **28% average FRQ score** (Across 12 scored free-response attempts for this unit.)
- **6.11: Integrating Using Integration by Parts (BC Only)**: 59% MCQ miss rate across 343 attempts. Review Integrating Using Integration by Parts (BC Only) with attention to how the concept appears in AP-style source and evidence questions.
- **6.10: Integrating Functions Using Long Division and Completing the Square**: 56% MCQ miss rate across 241 attempts. Review Integrating Functions Using Long Division and Completing the Square with attention to how the concept appears in AP-style source and evidence questions.
- **6.14: Selecting Techniques for Antidifferentiation**: 51% MCQ miss rate across 1974 attempts. Review Selecting Techniques for Antidifferentiation with attention to how the concept appears in AP-style source and evidence questions.
- **6.3: Riemann Sums, Summation Notation, and Definite Integral Notation**: 48% MCQ miss rate across 746 attempts. Review Riemann Sums, Summation Notation, and Definite Integral Notation with attention to how the concept appears in AP-style source and evidence questions.

## Review Notes

### 6.1: Accumulation of Change and Area

The area between a rate-of-change function and the x-axis represents accumulated change. When the rate is positive, accumulation is positive; when negative, accumulation is negative. For simple shapes (rectangles, triangles, semicircles), you can compute the area geometrically without an antiderivative. Units for accumulated change equal the rate unit multiplied by the input unit (for example, miles per hour times hours equals miles).

- **Accumulated change**: The total net change in a quantity over an interval, found as the signed area between the rate function and the x-axis.
- **Signed area**: Area counted as positive when the rate function is above the x-axis and negative when below.
- **Geometric evaluation**: Using area formulas for triangles, rectangles, and semicircles to compute a definite integral exactly when the graph forms those shapes.
- **Units of accumulation**: Rate unit times input unit; for example, (gallons per minute)(minutes) = gallons.

**Checkpoint:** Given a graph of a velocity function, can you find the net displacement over an interval using only geometry, and state the correct units?

### 6.2-6.3: Riemann Sums and Definite Integral Notation

When geometry is not enough, Riemann sums approximate the definite integral by summing products of function values and subinterval widths. Left, right, midpoint, and trapezoidal sums each have predictable over- or underestimate behavior based on whether the function is increasing, decreasing, concave up, or concave down. As the maximum subinterval width approaches zero, the Riemann sum limit equals the definite integral, written as the integral from a to b of f(x) dx.

- **Left Riemann sum**: Uses the left endpoint of each subinterval as the rectangle height; overestimates when f is decreasing, underestimates when increasing.
- **Right Riemann sum**: Uses the right endpoint; overestimates when f is increasing, underestimates when decreasing.
- **Trapezoidal sum**: Averages left and right endpoint values; overestimates when f is concave up, underestimates when concave down.
- **Definite integral as limit**: The integral from a to b of f(x) dx equals the limit of the Riemann sum as the maximum subinterval width approaches zero.
- **Summation notation**: A Riemann sum is written as the sum from i=1 to n of f(x_i*) times delta x_i, where x_i* is a sample point in the ith subinterval.

**Checkpoint:** Given a table of values with unequal subintervals, can you compute a left, right, and trapezoidal Riemann sum and identify which is an overestimate?

Method | Height used | Overestimates when | Underestimates when
--- | --- | --- | ---
Left sum | Left endpoint | f is decreasing | f is increasing
Right sum | Right endpoint | f is increasing | f is decreasing
Midpoint sum | Midpoint of subinterval | f is concave down | f is concave up
Trapezoidal sum | Average of endpoints | f is concave up | f is concave down

### 6.4-6.5: Accumulation Functions and FTC Part 1

An accumulation function g(x) = integral from a to x of f(t) dt defines a new function whose output is the net signed area under f from a to x. By FTC Part 1, g'(x) = f(x). This means you read the behavior of g directly from the graph of f: g increases where f is positive, decreases where f is negative, has a local extremum where f changes sign, and changes concavity where f changes from increasing to decreasing. When the upper limit is a composite function h(x), apply the chain rule: d/dx of the integral from a to h(x) of f(t) dt equals f(h(x)) times h'(x).

- **Accumulation function**: g(x) = integral from a to x of f(t) dt; g(a) = 0 and g'(x) = f(x).
- **FTC Part 1**: d/dx of the integral from a to x of f(t) dt = f(x), provided f is continuous on the interval.
- **Chain rule with variable limits**: d/dx of the integral from a to h(x) of f(t) dt = f(h(x)) times h'(x).
- **Increasing/decreasing of g**: g increases on intervals where f(x) > 0 and decreases where f(x) < 0.
- **Concavity of g**: g is concave up where f is increasing (g'' = f' > 0) and concave down where f is decreasing.

**Checkpoint:** Given a graph of f, can you sketch g(x) = integral from 0 to x of f(t) dt, label its local extrema, and find g'(3) without computing any antiderivative?

### 6.6-6.7: Properties of Definite Integrals and FTC Part 2

Definite integral properties let you manipulate and evaluate integrals algebraically. Reversing limits changes the sign; splitting an interval at an interior point uses additivity; constant multiples and sums distribute across the integral sign. FTC Part 2 gives the evaluation formula: the integral from a to b of f(x) dx = F(b) minus F(a), where F is any antiderivative of f. The constant of integration cancels in the subtraction, so you can use any convenient antiderivative.

- **Reversal of limits**: The integral from b to a of f(x) dx = negative of the integral from a to b of f(x) dx.
- **Additivity over intervals**: The integral from a to b plus the integral from b to c equals the integral from a to c.
- **FTC Part 2**: If F is an antiderivative of f on [a, b], then the integral from a to b of f(x) dx = F(b) minus F(a).
- **Discontinuities and integrability**: Definite integrals can be extended to functions with removable or jump discontinuities; the integral still exists.

**Checkpoint:** If you know the integral from 1 to 5 of f(x) dx = 10 and the integral from 1 to 3 of f(x) dx = 4, can you find the integral from 3 to 5 of f(x) dx using only properties?

### 6.8-6.9: Basic Antiderivative Rules and u-Substitution

Every differentiation rule has an integration counterpart. The reverse power rule gives the integral of x^n as x^(n+1)/(n+1) + C for n not equal to -1. Key antiderivatives to memorize include 1/x to ln|x| + C, e^x to e^x + C, sin x to -cos x + C, cos x to sin x + C, sec^2 x to tan x + C, 1/(1+x^2) to arctan x + C, and 1/sqrt(1-x^2) to arcsin x + C. U-substitution reverses the chain rule: set u = g(x), compute du = g'(x) dx, rewrite the integral in terms of u, integrate, then substitute back. For definite integrals, change the limits of integration to u-values instead of substituting back.

- **Reverse power rule**: Integral of x^n dx = x^(n+1)/(n+1) + C, valid for n not equal to -1.
- **Constant of integration**: The arbitrary constant C added to every indefinite integral, representing the family of all antiderivatives.
- **U-substitution**: Set u = inner function g(x), replace g'(x) dx with du, integrate in u, then back-substitute.
- **Changing limits under substitution**: For a definite integral, substitute the original limits into u = g(x) to get new limits; do not back-substitute.
- **Indefinite integral**: Written as the integral of f(x) dx = F(x) + C; represents all antiderivatives of f.

**Checkpoint:** Can you evaluate the integral of 2x times e^(x^2) dx using u-substitution, and correctly change the limits if the bounds are x = 0 to x = 2?

### 6.10: Long Division and Completing the Square

When the integrand is a rational function with numerator degree greater than or equal to denominator degree, use polynomial long division to rewrite it as a polynomial plus a proper fraction. When the denominator is an irreducible quadratic, complete the square to convert it to the form (x + a)^2 + b^2, which leads to an arctan antiderivative using the formula: integral of 1/(x^2 + a^2) dx = (1/a) arctan(x/a) + C. Both techniques rearrange the integrand into a form you already know how to integrate.

- **Polynomial long division**: Divides an improper rational function into a polynomial plus a proper fraction before integrating.
- **Completing the square**: Rewrites ax^2 + bx + c as a(x + b/2a)^2 + k to expose an arctan or arcsin antiderivative form.
- **Arctan antiderivative**: Integral of 1/(x^2 + a^2) dx = (1/a) arctan(x/a) + C, used after completing the square.
- **Rational functions**: Functions expressed as a ratio of two polynomials; integration technique depends on the degrees of numerator and denominator.

**Checkpoint:** Can you integrate (x^3 + 2x)/(x^2 + 1) dx by first applying long division, then integrating each resulting term?

### 6.11: Integration by Parts (BC Only)

Integration by parts applies when the integrand is a product of two functions and u-substitution does not work. The formula is: integral of u dv = uv minus the integral of v du. Choose u using the LIATE priority order (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) so that the new integral is simpler. Some integrals require repeated application; cyclic cases (such as e^x sin x) are solved by setting the repeated integral equal to a variable and solving algebraically.

- **Integration by parts formula**: Integral of u dv = uv minus the integral of v du; derived from the product rule for derivatives.
- **LIATE**: Mnemonic for choosing u: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential, in order of preference.
- **Cyclic integrals**: When repeated integration by parts returns the original integral, set it equal to I and solve algebraically.
- **Definite integrals by parts**: Evaluate the boundary term uv at the limits and subtract the integral of v du evaluated over the same limits.

**Checkpoint:** Can you evaluate the integral of x times e^x dx using integration by parts, and then evaluate the integral of e^x sin x dx using the cyclic method?

### 6.12: Linear Partial Fractions (BC Only)

Partial fraction decomposition rewrites a proper rational function whose denominator factors into distinct linear factors as a sum of simpler fractions, each of the form A/(ax + b). Each term integrates to a natural log expression. If the rational function is improper, apply polynomial long division first. Solve for the constants A, B, etc. by substituting the roots of each linear factor into the cleared equation or by matching coefficients.

- **Partial fraction decomposition**: Rewrites a rational function with distinct linear factors in the denominator as a sum of A/(ax + b) terms.
- **Solving for constants**: Substitute the root of each linear factor into the cleared equation to find each numerator constant directly.
- **Log antiderivative**: Integral of A/(ax + b) dx = (A/a) ln|ax + b| + C, the result after decomposition.

**Checkpoint:** Can you decompose 5/(x^2 - x - 6) into partial fractions, find the constants, and evaluate the resulting definite integral?

### 6.13: Improper Integrals (BC Only)

An improper integral has an infinite limit of integration or an integrand that is unbounded on the interval. Evaluate by replacing the problematic bound with a limit variable, integrating normally, then taking the limit. If the limit is a finite number, the integral converges; if the limit is infinite or does not exist, the integral diverges. When the integrand has a vertical asymptote inside the interval, split the integral at the asymptote and evaluate each piece separately.

- **Improper integral**: An integral with an infinite bound or an unbounded integrand; evaluated using a limit of a definite integral.
- **Convergence**: An improper integral converges when the limit of the definite integral is a finite number.
- **Divergence**: An improper integral diverges when the limit is infinite or does not exist.
- **Splitting at a discontinuity**: When the integrand has a vertical asymptote at an interior point c, write the integral as the sum of two improper integrals at c from the left and right.
- **Vertical asymptote**: A value where the integrand grows without bound; requires a one-sided limit when it falls within or at the boundary of the integration interval.

**Checkpoint:** Can you evaluate the integral from 1 to infinity of 1/x^2 dx and determine whether the integral from 0 to 1 of 1/sqrt(x) dx converges or diverges?

### 6.14: Selecting Antidifferentiation Techniques

Topic 6.14 is about recognition, not a new technique. Given any integrand, identify its structure and match it to the correct method: reverse power rule or known antiderivative for simple forms, u-substitution when an inner function and its derivative both appear, long division when the rational function is improper, completing the square for irreducible quadratic denominators, and (BC) integration by parts for products or isolated logarithms and inverse trig functions, partial fractions for factorable rational denominators, and limits for improper integrals.

- **Technique selection**: Identify the integrand structure first: polynomial, rational, product, composite, or with infinite bounds, then choose the matching method.
- **U-substitution signal**: Look for a composite function where the derivative of the inner function also appears as a factor in the integrand.
- **Integration by parts signal (BC)**: Look for a product of two different function types, especially when one factor simplifies upon differentiation (ln x, arctan x, x^n).
- **Partial fractions signal (BC)**: Look for a proper rational function with a factorable polynomial denominator.

**Checkpoint:** Given five integrals with different structures, can you correctly identify the technique for each before attempting any computation?

## Study Guides

- [6.1 Integration and Accumulation of Change](/ap-calc/unit-6/integration-accumulation-change/study-guide/NWRV9MaRJO4Eno32l5Xp)
- [6.11 Integrating Using Integration by Parts](/ap-calc/unit-6/integrating-using-integration-by-parts/study-guide/O4P3LchNoZnWElf8zETV)
- [6.12 Integrating Using Linear Partial Fractions](/ap-calc/unit-6/using-linear-partial-fractions/study-guide/VNjHMatlmaZoFyt7yLPH)
- [6.13 Evaluating Improper Integrals](/ap-calc/unit-6/evaluation-improper-integrals/study-guide/DRGur03jVQG3WXLiNyuz)
- [6.14 Selecting Techniques for Antidifferentiation (AB)](/ap-calc/unit-6/selecting-techniques-for-antidifferentiation/study-guide/oTEgMFOzDwdVR7OgZzk1)
- [6.2 Approximating Areas with Riemann Sums](/ap-calc/unit-6/approximating-areas-with-riemann-sums/study-guide/juN9YbvFYlJtpsMl)
- [6.3 Riemann Sums, Summation Notation, and Definite Integral Notation](/ap-calc/unit-6/riemann-sums-summation-notation-definite-integral-notation/study-guide/RnM03H2k6l3ewvxX)
- [6.4 The Fundamental Theorem of Calculus and Accumulation Functions](/ap-calc/unit-6/fundamental-theorem-calculus-accumulation-functions/study-guide/TyDEzN9M5wieA4Kw)
- [6.5 Interpreting the Behavior of Accumulation Functions Involving Area](/ap-calc/unit-6/interpreting-behavior-accumulation-functions-involving-area/study-guide/uVNoabC47nUgOdjr)
- [6.6 Applying Properties of Definite Integrals](/ap-calc/unit-6/applying-properties-definite-integrals/study-guide/lUbcVbDG5QVysAn9)
- [6.7 The Fundamental Theorem of Calculus and Definite Integrals](/ap-calc/unit-6/fundamental-theorem-calculus-definite-integrals/study-guide/fEGd7E9gbOH8EtCf)
- [6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation](/ap-calc/unit-6/finding-antiderivatives-indefinite-integrals-basic-rules-notation/study-guide/LfLFY5mlaTKUZXe3)
- [6.9 Integrating Using Substitution](/ap-calc/unit-6/integrating-using-substitution/study-guide/Oa1yaKFA8AzaadQq)
- [6.10 Integrating Functions Using Long Division and Completing the Square](/ap-calc/unit-6/integrating-functions-using-long-division-completing-square/study-guide/ju79RFY6f5aKWjFK)

## Practice Preview

### Multiple-choice practice

- **Stimulus-based practice question**: Practice 2 - Connecting Representations | Refer to the table. The integral and its antiderivative are shown using exponents A and B. Which of the following expressions is algebraically equivalent to the antiderivative shown and is correctly simplified for evaluating the definite integral?
- **AP-style practice question**: Practice 3 - Justification | To evaluate $$\int \frac{2x + 1}{x^2 + x + 3} \, dx$$ using substitution, a student must verify that which condition holds?
- **AP-style practice question**: Practice 3 - Justification | When evaluating $$\int_1^2 \sqrt{3x - 2} \, dx$$ using substitution $$u = 3x - 2$$, which of the following must be confirmed before proceeding?
- **AP-style practice question**: Practice 1 - Implementing Mathematical Processes | A particle moves along a line with velocity $$v(t) = 3t^2 - 12t + 9$$ meters per second for $$0 \leq t \leq 5$$. Which expression correctly represents the displacement of the particle over this time interval?
- **AP-style practice question**: Practice 2 - Connecting Representations | A function $$F(x) = \int_{3}^{x} \frac{1}{t} dt$$ is defined for $$x > 0$$. Which of the following correctly identifies $$F(x)$$ and $$F'(x)$$?
- **AP-style practice question**: Practice 2 - Connecting Representations | A particle's position function is defined by $$s(t) = \int_{0}^{t} v(\tau) d\tau + s_0$$, where $$v(\tau)$$ is its velocity. Which of the following scenarios shares the same underlying mathematical structure?

### FRQ practice

- **Water inflow rate and tank volume accumulation**: FRQs – No graphing calculator | Water inflow rate and tank volume accumulation
- **Rainwater flow approximation and basin accumulation**: FRQs – Graphing calculator required | Rainwater flow approximation and basin accumulation

## Key Terms

- **Antiderivative**: A function F whose derivative equals f. Finding antiderivatives is the core task of Unit 6; every integration technique produces an antiderivative.
- **Indefinite Integral**: Written as the integral of f(x) dx = F(x) + C; represents the entire family of antiderivatives of f, differing only by the constant C.
- **Constant of Integration**: The arbitrary constant C added to every indefinite integral. It cancels when evaluating a definite integral using F(b) minus F(a).
- **Riemann Sum**: An approximation of a definite integral computed by summing products of function values and subinterval widths over a partition of [a, b].
- **Left Riemann Sum**: A Riemann sum using the left endpoint of each subinterval as the function value; overestimates when f is decreasing, underestimates when increasing.
- **Right Riemann Sum**: A Riemann sum using the right endpoint of each subinterval; overestimates when f is increasing, underestimates when decreasing.
- **trapezoidal sum**: Approximates a definite integral by averaging the left and right endpoint values on each subinterval; overestimates when f is concave up.
- **Area under the curve**: The signed area between a function and the x-axis over an interval; equals the definite integral and represents accumulated change when f is a rate function.
- **Limits of Integration**: The values a and b in the integral from a to b of f(x) dx that specify the interval over which accumulation is measured.
- **U-Substitution**: An antidifferentiation technique that reverses the chain rule by substituting u = g(x) and du = g'(x) dx to simplify the integrand.
- **Completing the Square**: Rewriting a quadratic expression ax^2 + bx + c in the form a(x + h)^2 + k to convert a denominator into a form that yields an arctan antiderivative.
- **Improper Integral**: An integral with an infinite limit of integration or an unbounded integrand; evaluated as the limit of a definite integral and classified as convergent or divergent.
- **Rational Functions**: Functions expressed as a ratio of two polynomials; integrated using long division (if improper), completing the square, or partial fractions depending on the denominator structure.
- **Rate of Change**: The function f whose graph is used to measure accumulation; the area under a rate-of-change graph over an interval gives the net change in the original quantity.

## Common Mistakes

- **Forgetting to change limits in u-substitution for definite integrals**: When applying u-substitution to a definite integral, the limits of integration are x-values. You must substitute them into u = g(x) to get u-values before evaluating. Evaluating F(u) at the original x-limits gives the wrong answer.
- **Misreading over- and underestimate behavior of Riemann sums**: Over- and underestimate depend on both monotonicity and concavity depending on the method. Left and right sums depend on whether f is increasing or decreasing. Trapezoidal sums depend on concavity. Midpoint sums depend on concavity in the opposite direction from trapezoidal.
- **Dropping the negative sign when reversing limits**: The integral from b to a of f(x) dx equals the negative of the integral from a to b of f(x) dx. This sign flip is easy to miss when combining integrals over adjacent intervals or when applying FTC Part 2 with a lower variable limit.
- **Applying FTC Part 1 without the chain rule**: When the upper limit is h(x) rather than x, d/dx of the integral from a to h(x) of f(t) dt equals f(h(x)) times h'(x). Forgetting to multiply by h'(x) is one of the most common errors on accumulation function problems.
- **Omitting the constant of integration on indefinite integrals**: Every indefinite integral requires plus C. Omitting it is incorrect notation and loses points on free-response questions. The constant cancels in definite integral evaluation, but it must appear in all indefinite integral answers.

## Exam Connections

- **Evaluating and interpreting definite integrals in context**: AP Calculus free-response questions frequently present a rate function (velocity, flow rate, population growth) and ask you to compute a definite integral, interpret its meaning with correct units, or use FTC Part 2 to find a total accumulated value. Expect to apply both geometric evaluation from a graph and analytical evaluation using an antiderivative.
- **Differentiating accumulation functions**: Multiple-choice and free-response items regularly define a function as g(x) = integral from a to x of f(t) dt and ask for g'(x), g''(x), local extrema of g, or intervals where g is increasing or concave up. These require FTC Part 1, the chain rule for composite upper limits, and the ability to read f's graph to describe g's behavior without finding a formula.
- **Selecting and executing antidifferentiation techniques**: Both AB and BC exams include integrals that require technique identification before computation. AB items test u-substitution, basic rules, long division, and completing the square. BC items add integration by parts, partial fractions, and improper integrals. A common task pattern presents an integral without labeling the method, requiring you to recognize the structure and apply the correct procedure accurately.

## Final Review Checklist

- **Unit 6 final review checklist**: Use this checklist to confirm you can handle every major skill in Unit 6 before your exam.
- **Accumulation and Riemann sums**: Interpret area under a rate graph as accumulated change with correct units. Compute left, right, midpoint, and trapezoidal Riemann sums from a table or graph, and identify whether each is an over- or underestimate.
- **Definite integral notation and properties**: Translate between Riemann sum limit notation and definite integral notation. Apply reversal of limits, additivity over adjacent intervals, and constant multiple and sum rules to evaluate or simplify definite integrals.
- **Fundamental Theorem of Calculus**: Differentiate accumulation functions using FTC Part 1, including cases with a composite upper limit requiring the chain rule. Evaluate definite integrals using FTC Part 2 with the formula F(b) minus F(a).
- **Behavior of accumulation functions**: Given a graph of f, determine where g(x) = integral from a to x of f(t) dt is increasing, decreasing, concave up, concave down, and where it has local or absolute extrema.
- **Antidifferentiation techniques**: Apply the reverse power rule and all standard antiderivative formulas. Use u-substitution for indefinite and definite integrals, including changing limits. Apply long division and completing the square for rational function integrands.
- **BC-only techniques**: Apply integration by parts with correct u and dv selection, including cyclic cases. Decompose rational functions using linear partial fractions. Evaluate improper integrals using limits and determine convergence or divergence.
- **Technique selection**: Given an unfamiliar integrand, identify its structure and select the appropriate method without prompting. Practice mixed sets that include all techniques from 6.8-6.14.

## Study Plan

- **Step 1: Accumulation, Riemann sums, and notation (6.1-6.3)**: Start with the conceptual foundation. Read the topic guides for 6.1-6.3, practice computing all four Riemann sum types from a table with unequal subintervals, and practice translating a limit of a sum into definite integral notation. Confirm you can identify over- and underestimates using the comparison table.
- **Step 2: Fundamental Theorem of Calculus and accumulation functions (6.4-6.7)**: Work through the topic guides for 6.4-6.7. Practice differentiating accumulation functions with composite upper limits using the chain rule. Sketch g(x) from a graph of f without computing any antiderivative. Then practice evaluating definite integrals using F(b) minus F(a) and applying integral properties to combine known values.
- **Step 3: Basic antiderivatives and u-substitution (6.8-6.9)**: Memorize all standard antiderivative formulas from 6.8. Then drill u-substitution with both indefinite and definite integrals, paying close attention to limit changes. Use the topic guides for 6.8 and 6.9 and attempt practice questions that mix both skills.
- **Step 4: Long division, completing the square, and technique selection (6.10, 6.14 AB)**: Work through the topic guide for 6.10. Practice identifying when the numerator degree requires long division and when a quadratic denominator requires completing the square. Then use the 6.14 topic guide to practice mixed technique selection across all AB methods.
- **Step 5: BC-only techniques (6.11-6.14 BC)**: Work through topic guides for 6.11-6.13 in order. Practice integration by parts including cyclic cases, partial fraction decomposition with two or three linear factors, and improper integrals with both infinite bounds and interior asymptotes. Finish with mixed technique selection using the 6.14 topic guide to practice choosing among all BC methods.

## More Ways To Review

- [Topic study guides](/ap-calc/unit-6#topics)
- [FRQ practice](/ap-calc/frq-practice)
- [Cram archive videos](/cram-archives?subject=ap-calculus&unit=unit-6)
- [Key terms](/ap-calc/key-terms)

## FAQs

### What topics are covered in AP Calc Unit 6?

AP Calc Unit 6 covers 14 topics built around integration and accumulation of change. Key topics include Riemann Sums, the Fundamental Theorem of Calculus (Parts 1 and 2), accumulation functions, definite and indefinite integrals, u-substitution, and integration using long division. BC students also cover integration by parts, partial fractions, and improper integrals. Here's the full topic list:
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9 Integrating Using Substitution
- 6.10 Integrating Functions Using Long Division and Completing the Square
- 6.11 Integration by Parts (BC only)
- 6.12 Linear Partial Fractions (BC only)
- 6.13 Evaluating Improper Integrals (BC only)
- 6.14 Selecting Techniques for Antidifferentiation See [AP Calc Unit 6](/ap-calc/unit-6) for matched practice on every topic.

### How much of the AP Calc exam is Unit 6?

Unit 6 makes up 17-20% of the AP Calculus exam, making it one of the heaviest-weighted units on the test. It covers integration and accumulation of change, including Riemann Sums, the Fundamental Theorem of Calculus, antiderivatives, u-substitution, and several advanced techniques for BC students. That weight means roughly 1 in 5 exam points connects to this unit, so it's worth serious attention.

### What's on the AP Calc Unit 6 progress check (MCQ and FRQ)?

The AP Calc Unit 6 progress check in AP Classroom includes both MCQ and FRQ parts drawn from the unit's 14 topics. The MCQ section tests skills like setting up Riemann Sums, applying properties of definite integrals, and evaluating antiderivatives using substitution or basic rules. The FRQ part typically asks you to interpret accumulation functions, apply the Fundamental Theorem of Calculus, and select appropriate antidifferentiation techniques. BC students also see questions on integration by parts, partial fractions, and improper integrals in their progress check. Practicing these topics before the progress check at [AP Calc Unit 6](/ap-calc/unit-6) will help you spot which techniques you still need to sharpen.

### How do I practice AP Calc Unit 6 FRQs?

AP Calc Unit 6 FRQs most often pull from the Fundamental Theorem of Calculus, accumulation functions, and selecting antidifferentiation techniques, so those are the topics to prioritize. A typical FRQ asks you to evaluate a definite integral, interpret what an accumulation function represents in context, or justify behavior using area under a curve. To practice, work through released College Board FRQs that involve integration, write out every step of your reasoning (not just the answer), and check that your notation for definite and indefinite integrals is clean. BC students should also practice integration by parts and improper integrals in FRQ format. Find topic-aligned practice at [AP Calc Unit 6](/ap-calc/unit-6).

### Where can I find AP Calc Unit 6 practice questions?

The best place to find AP Calc Unit 6 practice questions, including multiple-choice and practice test sets, is [AP Calc Unit 6](/ap-calc/unit-6). That page has resources organized by topic, so you can target Riemann Sums, the Fundamental Theorem of Calculus, u-substitution, or any of the other 14 topics in this unit. For MCQ practice, focus on questions that ask you to evaluate definite integrals, interpret accumulation functions, or choose the right antidifferentiation technique. Released College Board exams are also a strong source for realistic practice test questions on integration.

### How should I study AP Calc Unit 6?

Start AP Calc Unit 6 by building a solid understanding of Riemann Sums and definite integral notation before moving to the Fundamental Theorem of Calculus, since later topics stack on those foundations. Then work through antiderivative rules, u-substitution, and long division in order, checking your understanding with practice problems after each topic. A concrete study plan: review one topic per session, do at least five practice problems per topic, and then take a timed MCQ set at the end of the unit to see which techniques still feel shaky. BC students should budget extra time for integration by parts, partial fractions, and improper integrals. Keep your notation tight throughout. Definite integrals with wrong bounds or missing dx are common point losses on the exam. Use [AP Calc Unit 6](/ap-calc/unit-6) to find topic-specific practice as you go.

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BC students also see questions on integration by parts, partial fractions, and improper integrals in their progress check. Practicing these topics before the progress check at <a href=\"/ap-calc/unit-6\">AP Calc Unit 6</a> will help you spot which techniques you still need to sharpen."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-6#how-do-i-practice-ap-calc-unit-6-frqs","name":"How do I practice AP Calc Unit 6 FRQs?","acceptedAnswer":{"@type":"Answer","text":"AP Calc Unit 6 FRQs most often pull from the Fundamental Theorem of Calculus, accumulation functions, and selecting antidifferentiation techniques, so those are the topics to prioritize. A typical FRQ asks you to evaluate a definite integral, interpret what an accumulation function represents in context, or justify behavior using area under a curve. To practice, work through released College Board FRQs that involve integration, write out every step of your reasoning (not just the answer), and check that your notation for definite and indefinite integrals is clean. BC students should also practice integration by parts and improper integrals in FRQ format. Find topic-aligned practice at <a href=\"/ap-calc/unit-6\">AP Calc Unit 6</a>."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-6#where-can-i-find-ap-calc-unit-6-practice-questions","name":"Where can I find AP Calc Unit 6 practice questions?","acceptedAnswer":{"@type":"Answer","text":"The best place to find AP Calc Unit 6 practice questions, including multiple-choice and practice test sets, is <a href=\"/ap-calc/unit-6\">AP Calc Unit 6</a>. That page has resources organized by topic, so you can target Riemann Sums, the Fundamental Theorem of Calculus, u-substitution, or any of the other 14 topics in this unit. For MCQ practice, focus on questions that ask you to evaluate definite integrals, interpret accumulation functions, or choose the right antidifferentiation technique. Released College Board exams are also a strong source for realistic practice test questions on integration."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-6#how-should-i-study-ap-calc-unit-6","name":"How should I study AP Calc Unit 6?","acceptedAnswer":{"@type":"Answer","text":"Start AP Calc Unit 6 by building a solid understanding of Riemann Sums and definite integral notation before moving to the Fundamental Theorem of Calculus, since later topics stack on those foundations. Then work through antiderivative rules, u-substitution, and long division in order, checking your understanding with practice problems after each topic. A concrete study plan: review one topic per session, do at least five practice problems per topic, and then take a timed MCQ set at the end of the unit to see which techniques still feel shaky. BC students should budget extra time for integration by parts, partial fractions, and improper integrals. Keep your notation tight throughout. Definite integrals with wrong bounds or missing dx are common point losses on the exam. Use <a href=\"/ap-calc/unit-6\">AP Calc Unit 6</a> to find topic-specific practice as you go."}}]}
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