AP Calculus ABAP Exams

AP Calculus AB Unit 6 Review: The Most-Tested Riemann Sum and FTC Questions

Sigma Prep

Math Instructor · 10+ Years Experience

October 24, 2026·14 min read

Key takeaways

Unit weight: 61 of 315 official multiple choice questions (19.4%), the most official multiple choice of any unit across all 17 exams, and 13.5% of the free-response points.
Top types: Riemann and trapezoidal sums from a table, the behavior of $g(x) = \int_a^x f(t)\,dt$ , and the derivative of an accumulation function.
Practice: table sums with uneven widths. One was on the free response of every exam from 2012 to 2026.

Unit 6 has more official multiple choice questions than any other AP Calculus AB unit. College Board’s Course and Exam Description (the CED) gives Unit 6 15–20% of the multiple choice. Across all 17 official exams from 1969 to 2018 it has 159 of the 719 in-scope multiple choice questions (22.1%), the most of any unit. On the 7 official practice exams from 2012–2018 it was 61 of 315 (19.4%), between 8 and 10 per exam. On the 42-question section that starts in May 2027, that rate is about 8 questions.

It counts on the free response too. Unit 6 earned 13.5% of the points on the 2012–2026 exams, and 39 of the 86 free-response questions since 2012 had a Unit 6 part, on every one of the 15 exams. The common ones:

Riemann and trapezoidal sums from a table (15 questions, one on every exam from 2012 to 2026), usually in the calculator part: approximate the integral of a rate, with units.
Integrals from the geometry of a graph (14 questions on 14 exams): areas of triangles, rectangles and semicircles, with area below the axis counted negative.
The derivative of an accumulation function (12 questions on 10 exams): $g(x) = \int_a^x f(t)\,dt$ has $g' = f$ .
Net change (7 questions): $f(b) = f(a) + \int_a^b f'(x)\,dx$ .
Substitution (8 questions on 7 exams), mostly inside a larger question.

The 12 most-tested Unit 6 question types

Ranked by share of the whole exam. The twelve add up to about 12.5% of the exam, out of 14.8% for all of Unit 6. Eight smaller types follow them.

1. Riemann and trapezoidal sums from a table

Each subinterval contributes its own width times a height: the left value, the right value, or the average of the two for a trapezoid. Tables on the exam often have uneven widths.

Topic 6.22.3% of the exam5 of 7 practice exams (5 questions)15 FRQs since 2012, one on every exam

Try it

Water flows into a tank at

R(t)

gallons per minute. Selected values are in the table.

$t$ (minutes)	0	2	5	9
$R(t)$ (gal/min)	4	7	6	10

Use a trapezoidal sum with these three subintervals to approximate

\int_0^9 R(t)\,dt

Solution

$2 \cdot \frac{4 + 7}{2} + 3 \cdot \frac{7 + 6}{2} + 4 \cdot \frac{6 + 10}{2} = 11 + 19.5 + 32 = 62.5$ gallons. (The left sum is $2 \cdot 4 + 3 \cdot 7 + 4 \cdot 6 = 53$ gallons.)

Common mistake

using the wrong values or the wrong widths: a left sum uses the first value of each subinterval and a right sum the last, each times its own width, and the subintervals in a table are often uneven.

Practice: Riemann Sums From a Table worksheet · AP Unit 6 worksheet

2. Analyze $g(x) = \int_a^x f(t)\,dt$

$g' = f$ and $g'' = f'$ . So $g$ increases where $f$ is positive, has a relative maximum where $f$ changes from positive to negative, and is concave up where $f$ increases.

Topic 6.51.6% of the exam4 of 7 practice exams (5 questions)7 FRQs since 2012

Try it

Let

g(x) = \int_0^x (t^2 - 4t + 3)\,dt

. Where does

g

have relative extrema and a point of inflection?

Solution

$g'(x) = (x - 1)(x - 3)$ , which changes from positive to negative at $x = 1$ (relative maximum) and from negative to positive at $x = 3$ (relative minimum). $g''(x) = 2x - 4$ changes sign at $x = 2$ , the point of inflection.

Common mistake

mixing up

g

g' = f

and

g'' = f'

g

increases where

f > 0

(not where

f

increases), and its graph is concave up where

f

increases.

Practice: Accumulation Functions worksheet · AP Unit 6 worksheet

3. The derivative of an accumulation function (FTC)

$\frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f(u(x)) \cdot u'(x)$ . With $x$ as the upper limit, the chain-rule factor is 1.

Topic 6.41.5% of the exam4 of 7 practice exams (5 questions)12 FRQs since 2012

Try it

F(x) = \int_1^{x^3} \frac{1}{1 + t^2}\,dt

. Find

F'(2)

Solution

$F'(x) = \frac{3x^2}{1 + x^6}$ , so $F'(2) = \frac{12}{65}$ .

Common mistake

leaving off the chain-rule factor

u'(x)

: here that gives

\frac{1}{65}

Practice: The Second Fundamental Theorem worksheet · AP Unit 6 worksheet

4. Net change: $f(b)$ from $f(a)$ and the integral of $f'$

$f(b) = f(a) + \int_a^b f'(x)\,dx$ . The integral is the change; the answer is the starting value plus the change. $f'$ may be a formula, a graph or a table.

Topic 6.71.4% of the exam4 of 7 practice exams (6 questions)7 FRQs since 2012

Try it

f'(x) = \cos(x^2)

and

f(0) = 2

. What is

f(1)

Solution

$\cos(x^2)$ has no elementary antiderivative, so use the calculator’s integral: $f(1) = 2 + \int_0^1 \cos(x^2)\,dx \approx 2 + 0.905 = 2.905$ .

Common mistake

giving only the integral, the change (0.905 here), and forgetting to add the starting value

f(a)

. Going backward (

b < a

) the change is subtracted, and area below the axis lowers

f

Practice: Net Change worksheet · AP Unit 6 worksheet

5. Definite integrals from the geometry of a graph

Split the region into triangles, rectangles and circle pieces. Area above the $x$ -axis counts positive, area below counts negative.

Topic 6.61.1% of the exam3 of 7 practice exams (4 questions)14 FRQs since 2012

Try it

The graph of

f

is shown: a line segment from

(0, 2)

(2, 0)

, then a semicircle of radius 2 below the

x

-axis from

x = 2

x = 6

Find

\int_0^6 f(x)\,dx

Solution

the triangle gives 2 and the semicircle $-\frac{1}{2}\pi \cdot 2^2 = -2\pi$ , so the integral is $2 - 2\pi \approx -4.283$ .

Common mistake

adding areas below the

x

-axis instead of subtracting them, or using the wrong fraction of a circle: a semicircle of radius

r

has area

\frac{1}{2}\pi r^2

, a quarter circle

\frac{1}{4}\pi r^2

Practice: Definite Integrals worksheet · AP Unit 6 worksheet

6. Rewrite an integral in terms of $u$

Replace every $x$ , including $dx$ , and for a definite integral change the limits to values of $u$ . The question often asks only for the equivalent integral.

Topic 6.91.0% of the exam4 of 7 practice exams (6 questions)3 FRQs since 2012 (2014, 2016, 2020)

Try it

Using

u = x^2 + 1

, write

\int_0^2 x(x^2 + 1)^3\,dx

in terms of

u

and evaluate it.

Solution

$du = 2x\,dx$ , and $x = 0$ , 2 give $u = 1$ , 5. The integral is $\frac{1}{2}\int_1^5 u^3\,du = \frac{1}{2} \cdot \frac{625 - 1}{4} = 78$ .

Common mistake

changing the integrand to

u

but keeping the

x

-limits (

\frac{1}{2}\int_0^2 u^3\,du = 2

here), or forgetting the constant from

du

Practice: Integration by Substitution worksheet · AP Unit 6 worksheet

7. Evaluate a definite integral with the FTC

Find an antiderivative $F$ and compute $F(b) - F(a)$ .

Topic 6.70.8% of the exam2 of 7 practice exams (3 questions)3 FRQs since 2012 (2018, 2019, 2023)

Try it

Evaluate

\int_1^4 \left(3\sqrt{x} + \frac{2}{x^2}\right) dx

Solution

an antiderivative is $2x^{3/2} - \frac{2}{x}$ . $\left(16 - \frac{1}{2}\right) - (2 - 2) = \frac{31}{2}$ .

Common mistake

evaluating the antiderivative at the upper limit only, or subtracting in the wrong order:

\int_a^b f(x)\,dx = F(b) - F(a)

for an antiderivative

F

, never values of

f

itself.

Practice: Definite Integrals worksheet · AP Unit 6 worksheet

8. u-substitution (indefinite)

Look for a function whose derivative is also a factor, up to a constant. Set $u$ to the inside function and adjust for the constant in $du$ .

Topic 6.90.8% of the exam3 of 7 practice exams (4 questions)never the main type of a free-response part

Try it

Find

\int 6x(3x^2 - 1)^4\,dx

Solution

with $u = 3x^2 - 1$ , $du = 6x\,dx$ , so the integral is $\int u^4\,du = \frac{(3x^2 - 1)^5}{5} + C$ .

Common mistake

dropping the constant that

du

needs. With

u = x^3 + 5

du = 3x^2\,dx

, so

\int x^2(x^3 + 5)^6\,dx = \frac{1}{21}(x^3 + 5)^7 + C

, not

\frac{1}{7}(x^3 + 5)^7 + C

Practice: Integration by Substitution worksheet · AP Unit 6 worksheet

9. Combine given values of integrals

Use the properties: integrals over adjacent intervals add, reversing the limits changes the sign, constants factor out, and $\int_a^b c\,dx = c(b - a)$ .

Topic 6.60.6% of the exam5 of 7 practice exams (5 questions)1 FRQ since 2012 (2019)

Try it

\int_1^5 f(x)\,dx = 8

and

\int_3^5 f(x)\,dx = -2

. Find

\int_1^3 (2f(x) + 1)\,dx

Solution

$\int_1^3 f = 8 - (-2) = 10$ , so the integral is $2 \cdot 10 + 1 \cdot (3 - 1) = 22$ .

Common mistake

forgetting that reversing the limits changes the sign, or integrating a constant term

c

c

instead of

c(b - a)

(that gives 21 here).

Practice: Properties of Definite Integrals worksheet · AP Unit 6 worksheet

10. Basic antiderivatives

Reverse the derivative rules: the power rule, $\frac{1}{x}$ , $e^x$ , the trig functions, and the inverse trig forms.

Topic 6.80.5% of the exam3 of 7 practice exams (3 questions)never the main type of a free-response part

Try it

Find

\int \left(4x^3 - \frac{3}{x} + \sec^2 x\right) dx

Solution

$x^4 - 3\ln|x| + \tan x + C$ .

Common mistake

differentiating instead of integrating, or forgetting to divide by the new exponent:

\int x^r\,dx = \frac{x^{r+1}}{r + 1} + C

. A constant term

k

integrates to

kx

, and

\int \frac{1}{x^2}\,dx = -\frac{1}{x} + C

, not a logarithm.

Practice: Integration by the Power Rule worksheet · AP Unit 6 worksheet

11. u-substitution (definite)

Substitute, change the limits to $u$ -values, and evaluate in $u$ ; or find the antiderivative in $x$ and use the original limits. Not a mix.

Topic 6.90.5% of the exam2 of 7 practice exams (2 questions)5 FRQs since 2012, 4 of them as part of a larger question

Try it

Evaluate

\int_0^1 xe^{x^2}\,dx

Solution

with $u = x^2$ , $du = 2x\,dx$ and the limits stay 0 and 1: $\frac{1}{2}\int_0^1 e^u\,du = \frac{e - 1}{2}$ .

Common mistake

forgetting the constant from

du

(that gives

e - 1

here) or evaluating at the old

x

-limits after changing to

u

Practice: Integration by Substitution worksheet · AP Unit 6 worksheet

12. Integrals of piecewise and absolute-value functions

Split the integral where the formula changes, or where the expression inside the absolute value changes sign.

Topic 6.60.4% of the exam2 of 7 practice exams (2 questions)2 FRQs since 2012 (2018, 2026)

Try it

Evaluate

\int_{-1}^3 |x - 1|\,dx

Solution

split at $x = 1$ . Each side is a triangle with base 2 and height 2, area 2, so the integral is 4.

Common mistake

integrating

|g(x)|

without splitting where

g(x) = 0

(here

\int_{-1}^3 (x - 1)\,dx = 0

), or using one formula of a piecewise function over the whole interval.

Practice: Definite Integrals worksheet · AP Unit 6 worksheet

Eight more Unit 6 types, lower priority

13. Integrals that give a natural log

When the numerator is the derivative of the denominator, up to a constant, the integral is a logarithm.

Topic 6.90.4% of the exam3 of 7 practice exams (3 questions)1 FRQ since 2012 (2019), as part of a larger question

Try it

Find

\int \frac{x}{x^2 + 9}\,dx

Solution

with $u = x^2 + 9$ , $du = 2x\,dx$ , so the integral is $\frac{1}{2}\ln(x^2 + 9) + C$ . No absolute value is needed, since $x^2 + 9 > 0$ .

Common mistake

using the power rule on

u^{-1}

, or losing the constant from

du

(

\ln(x^2 + 9) + C

). For

\int \frac{dx}{5 - 3x}

the sign of the inside’s derivative matters:

-\frac{1}{3}\ln|5 - 3x| + C

Practice: Integration by Substitution worksheet · AP Unit 6 worksheet

14. Antiderivative with an initial condition

Find the general antiderivative with $+ C$ , then use the given point to solve for $C$ .

Topic 6.80.4% of the exam2 of 7 practice exams (2 questions)1 FRQ since 2012 (2022), as part of a larger question

Try it

f'(x) = 3x^2 - 4x

and

f(1) = 5

. Find

f(x)

Solution

$f(x) = x^3 - 2x^2 + C$ , and $f(1) = -1 + C = 5$ gives $C = 6$ : $f(x) = x^3 - 2x^2 + 6$ .

Common mistake

using the initial value as the constant (

C = 5

here). Solve

C = y_0 - F(x_0)

;

C = y_0

only when

F(x_0) = 0

Practice: Antiderivatives With Initial Conditions worksheet · AP Unit 6 worksheet

15. Integral properties: symmetry, shifts and bounds

An odd function integrates to 0 over $[-a, a]$ , an even one gives twice the integral over $[0, a]$ . If $m \le f \le M$ on $[a, b]$ , the integral is between $m(b - a)$ and $M(b - a)$ .

Topic 6.60.3% of the exam1 of 7 practice exams (1 question)no FRQs since 2012 (the last was in 2006)

Try it

f

is odd and continuous. Find

\int_{-3}^3 (f(x) + 2)\,dx

Solution

the $f$ part is 0 by symmetry and the constant gives $2 \cdot 6 = 12$ .

Common mistake

mixing up the symmetry rules (an odd function cancels on

[-a, a]

, an even one doubles), or treating

\int f(x)g(x)\,dx

\int f(x)\,dx \cdot \int g(x)\,dx

Practice: Properties of Definite Integrals worksheet · AP Unit 6 worksheet

16. Over- or underestimate?

Left and right sums depend on whether $f$ is increasing or decreasing. Trapezoidal and midpoint sums depend on concavity, and they err in opposite directions.

Topic 6.20.3% of the exam1 of 7 practice exams (1 question)4 FRQs since 2012 (2012, 2016, 2017, 2021), all in the calculator part

Try it

f

is increasing and concave down on

[0, 4]

. Which of the left, right, trapezoidal and midpoint sums underestimate

\int_0^4 f(x)\,dx

Solution

the left sum ( $f$ increasing) and the trapezoidal sum (concave down: each chord lies below the curve). The right and midpoint sums overestimate.

Common mistake

deciding every sum from the same property. On the free response, the reason (“

f

is increasing” or “the graph is concave down”) earns the point.

Practice: Riemann Sums worksheet · AP Unit 6 worksheet

17. Riemann or trapezoidal sum from a function or graph

With $n$ equal subintervals on $[a, b]$ , each has width $\frac{b - a}{n}$ .

Topic 6.20.2% of the exam1 of 7 practice exams (1 question)no FRQs since 2012 (the last was in 2002)

Try it

Find the right Riemann sum for

\int_0^4 x^2\,dx

with 4 equal subintervals.

Solution

width 1, heights at $x = 1$ , 2, 3, 4: $1 + 4 + 9 + 16 = 30$ . (The left sum is 14 and the integral is $\frac{64}{3}$ .)

Common mistake

using the wrong width: every height is multiplied by

\frac{b - a}{n}

Practice: Riemann Sums worksheet · AP Unit 6 worksheet

18. Long division and completing the square

If the top’s degree is at least the bottom’s, divide first. For a constant over a quadratic with no real roots, complete the square and use arctan.

Topic 6.100.2% of the exam1 of 7 practice exams (1 question)never on the free response

Try it

Find

\int \frac{dx}{x^2 - 4x + 13}

Solution

$x^2 - 4x + 13 = (x - 2)^2 + 9$ , so the integral is $\frac{1}{3}\arctan\left(\frac{x - 2}{3}\right) + C$ .

Common mistake

writing a natural log: the numerator is not the derivative of the denominator, so complete the square instead.

Practice: Inverse Trig Integrals worksheet · AP Unit 6 worksheet

19. Riemann sum notation to an integral

In the limit of a Riemann sum, the width $\frac{b - a}{n}$ becomes $dx$ , and the points $a + k\frac{b - a}{n}$ run from $a$ to $b$ .

Topic 6.30.1% of the exam1 of 7 practice exams (1 question)never on the free response

Try it

Write

\lim\limits_{n \to \infty} \sum_{k=1}^{n} \frac{2}{n}\left(1 + \frac{2k}{n}\right)^3

as an integral and evaluate it.

Solution

the width is $\frac{2}{n}$ and the points run from 1 to 3: $\int_1^3 x^3\,dx = \frac{81 - 1}{4} = 20$ .

Common mistake

reading the limits off the sum wrongly, such as

\int_0^2 x^3\,dx = 4

here.

Practice: Area by the Limit Definition worksheet · AP Unit 6 worksheet

20. Accumulation as the area under a rate graph

The amount accumulated is the area between the rate graph and the axis, with units of rate times time.

Topic 6.10.1% of the exam1 of 7 practice exams (1 question)never the main type of a free-response part

Try it

Water flows into a tank at 3 liters per minute for 4 minutes, then the rate drops steadily to 0 at

t = 6

. How much water flows in?

Solution

a rectangle, $3 \cdot 4 = 12$ , plus a triangle, $\frac{1}{2} \cdot 2 \cdot 3 = 3$ : 15 liters.

Common mistake

giving a rate as the amount, or multiplying one rate by the elapsed time (

3 \cdot 6 = 18

liters here).

Practice: Net Change worksheet · AP Unit 6 worksheet

What’s rising and fading in Unit 6

We compare each type’s share of recent exams (multiple choice 2012–2018, free response 2012–2026) with its share across all years: Rising is at least double, Fading under half. The movement is toward tables, graphs and accumulation, and away from plain integral computation.

Rising: rewriting an integral in terms of $u$ . 1.38% of recent exams against 0.66% across all years, and 6 of its 8 official multiple choice questions are on the 2012–2018 practice exams. Teacher material confirms the rise. Its free-response share went the other way, from 1.6% of the points (2012–2018) to 0.1% (2019–2026).
Up: Riemann sums from a table. 2.79% of recent exams against 1.59% across all years, the biggest gain in the unit. Net change, $f(b) = f(a) + \int f'$ , is up too: 1.87% against 1.21%.
Up on the newest free response: the derivative of an accumulation function. 0.8% of the free-response points in 2012–2018, 2.2% in 2019–2026. Counted by questions, it was in 12% of the free-response questions from 1998 to 2018 and 20% from 2019 to 2026.
Fading: evaluating a definite integral with the FTC (0.67% against 1.37%) and an antiderivative with an initial condition (0.32% against 0.64%), both confirmed by teacher material, and integral properties such as symmetry and bounds (0.16% against 0.56%, last official multiple choice question in 2012).

Unit 6 as a whole went from 14.0% to 13.0% of the free-response points. One question is about 2.3% of a window’s points, so that is less than one question.

In the CED, rarely asked in multiple choice

Selecting techniques for antidifferentiation (topic 6.14) has never been the main topic of an official question.
Exploring accumulations of change (topic 6.1) has 2 official multiple choice questions, and Riemann sum notation (6.3) 2. Neither has been the main type of a free-response part.
Long division and completing the square (topic 6.10) has 3 official multiple choice questions and has never been the main type of a free-response part.
Basic antiderivatives and indefinite u-substitution have 11 and 16 official multiple choice questions, but neither has been the main type of a free-response part since 1998.
Integration by parts (topic 6.11) is BC only. It had 4 multiple choice questions on the old AB exams (1985–1997), none since.

So the free response in Unit 6 is about tables, graphs and accumulation; the antiderivative techniques are tested mostly in multiple choice.

What students over- and under-practice in Unit 6

We tagged 10,263 multiple choice questions from teacher worksheets, review packets and prep books (reposted official questions and non-AB content removed). Unit 6 is 20.3% of them, against 22.1% of the 719 in-scope official multiple choice questions from 1969–2018, so the unit as a whole gets about its share. Per type:

Question type	Practice vs official share	Ratio
Antiderivative with an initial condition	0.70% vs 0.42%	1.7× over
Riemann sum from a function or graph	0.68% vs 0.42%	1.6× over
u-substitution (indefinite)	3.12% vs 2.23%	1.4× over
Integrals that give ln	1.33% vs 1.11%	1.2× over
Derivative of an accumulation function	1.70% vs 1.53%	1.1×, about even
Analyze g(x) = ∫ f	1.37% vs 1.25%	1.1×, about even
Basic antiderivatives	1.67% vs 1.53%	1.1×, about even
Integrals from graph geometry	0.90% vs 0.97%	0.9×, about even
u-substitution (definite)	1.25% vs 1.39%	0.9×, about even
Riemann sums from a table	0.90% vs 1.11%	0.8× under
Definite integral by the FTC	1.83% vs 2.36%	0.8× under
Rewrite an integral in terms of u	0.80% vs 1.11%	0.7× under
Over- or underestimate	0.35% vs 0.56%	0.6× under
Net change from f(a) and ∫ f′	0.99% vs 1.67%	0.6× under
Piecewise and absolute-value integrals	0.53% vs 0.97%	0.5× under
Combine given values of integrals	0.74% vs 1.39%	0.5× under
Accumulation from a rate graph	0.06% vs 0.28%	0.2× under

Official counts per type are small (2 to 17 questions), so read the ratios as directions.

Practice leans on antiderivative mechanics: indefinite u-substitution is the most common Unit 6 practice question by far, at 1.4 times its official share.
The reasoning types are under-practiced: combining given values (0.5×), net change (0.6×) and over- or underestimates (0.6×), all exam staples. Combining given values is mostly a calculator-section question (7 of 10).
Accumulation from a rate graph is nearly absent (6 practice questions in our whole set), though accumulating a rate in context is the biggest theme of the free response.

Calculator note

Unit 6 is mostly no-calculator. Of the 61 Unit 6 multiple choice questions on the 2012–2018 practice exams, 44 were in the no-calculator part and 17 in the calculator part. The calculator part had 4 of the 5 questions on combining given values of integrals and 3 of the 6 net-change questions. Every table Riemann sum (5), derivative of an accumulation function (5), indefinite substitution (4), basic antiderivative (3) and log integral (3) was no-calculator.

Across all 17 official exams, 41 of the 159 Unit 6 multiple choice questions were in the calculator section: 24 give a formula, 10 a graph and 6 a table. Combining given values of integrals had 7 of its 10 questions there, though it needs only arithmetic.

On free response, 11 of the 39 questions with a Unit 6 part allowed a calculator (counting 2020, when a calculator was allowed throughout): 9 of the 15 table-sum questions, but only 1 of the 14 graph-geometry questions and 1 of the 7 net-change questions.

So know how to compute a definite integral on your calculator, for an amount from a rate model or a net change with an $f'$ like $\cos(x^2)$ , and give answers to three decimal places. Do the substitutions, the graph areas and the accumulation functions by hand.

How we counted

Every number here comes from the public official AP Calculus AB exams: 760 multiple choice questions (1969–2018) and 416 free-response questions (1969–2026), each tagged to its CED topic and question type. Every example problem is our own.

Practice Unit 6

Our free AP Calculus AB Unit 6 worksheet has 20 original multiple choice questions on Riemann sums, accumulation functions, the Fundamental Theorem of Calculus and integration by substitution, one on each of the 20 types above (choosing a technique, topic 6.14, is left out). Part A has 15 questions without a calculator and Part B has 5 with one. Part B has a table sum, an over- or underestimate, an amount from a rate, and two questions that combine given values or the symmetry of an integral with an integral only a calculator can evaluate. Two or three questions in each version need a graphing calculator, with answers to three decimal places. It comes with an answer key and a printable PDF.

For free response, the AP Calculus AB Unit 6 FRQ set has 4 multi-part questions in the exam’s form, each scored out of 9 points, with a scoring guide and a PDF. Both make good practice before the Unit 6 Progress Check in AP Classroom.

Common Questions

The CED gives Unit 6 15–20% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 61 of 315 multiple choice questions (19.4%), 8 to 10 per exam, and across all 17 official exams it has 159 multiple choice questions, more than any other unit. In free response it earned 13.5% of the points on the 2012–2026 exams, with a Unit 6 part on every exam: table Riemann sums, integrals from a graph, accumulation functions and net change.

Every year. A Riemann or trapezoidal sum from a table was on all 15 exams from 2012 to 2026, in 9 of them in the calculator part. Expect uneven subintervals, units, and often a follow-up: what the integral means in context, or whether the sum over- or underestimates it.

Two ways. The derivative of an accumulation function, d/dx of the integral from a to x of f(t) dt = f(x), with the chain rule when the upper limit is a function of x: 11 official multiple choice questions and 12 free-response questions since 2012, on 10 of the 15 exams. And net change, f(b) = f(a) + the integral of f′ from a to b: 12 official multiple choice questions, half of them in the calculator section, and 7 free-response questions since 2012.

Yes, mostly in multiple choice. Integrating by substitution (topic 6.9) has 42 official multiple choice questions, on all 17 exams, the most of any Unit 6 topic. Rewriting an integral in terms of u, with new limits, is rising: 6 of its 8 official questions are on the 2012–2018 practice exams. On the free response, substitution was in 8 questions since 2012, usually as one step of a larger question.

Rarely in multiple choice. Of the 61 Unit 6 multiple choice questions on the 2012–2018 practice exams, 17 were in the calculator part, and many of those, such as combining given values of integrals, need only arithmetic. Use the calculator's definite integral for an amount from a rate model or a net change with an f′ you cannot integrate by hand, such as cos(x²), and give answers to three decimal places. On the free response, 9 of the 15 table Riemann sum questions since 2012 were in the calculator part.

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Practice this unit: free AP Calculus AB Unit 6 worksheet

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AP Calculus AB Unit 6 Review: The Most-Tested Riemann Sum and FTC Questions

The 12 most-tested Unit 6 question types

1. Riemann and trapezoidal sums from a table

2. Analyze g(x)=∫axf(t) dtg(x) = \int_a^x f(t)\,dtg(x)=∫ax​f(t)dt

3. The derivative of an accumulation function (FTC)

4. Net change: f(b)f(b)f(b) from f(a)f(a)f(a) and the integral of f′f'f′

5. Definite integrals from the geometry of a graph

6. Rewrite an integral in terms of uuu

7. Evaluate a definite integral with the FTC

8. u-substitution (indefinite)

9. Combine given values of integrals

10. Basic antiderivatives

11. u-substitution (definite)

12. Integrals of piecewise and absolute-value functions

Eight more Unit 6 types, lower priority

13. Integrals that give a natural log

14. Antiderivative with an initial condition

15. Integral properties: symmetry, shifts and bounds

16. Over- or underestimate?

17. Riemann or trapezoidal sum from a function or graph

18. Long division and completing the square

19. Riemann sum notation to an integral

20. Accumulation as the area under a rate graph

What’s rising and fading in Unit 6

In the CED, rarely asked in multiple choice

What students over- and under-practice in Unit 6

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How we counted

Practice Unit 6

Common Questions

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Practice this unit: free AP Calculus AB Unit 6 worksheet

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AP Calculus AB Review: The Most-Tested Questions, Unit by Unit

AP Calculus AB Unit 1 Review: The Most-Tested Limits and Continuity Questions

AP Calculus AB Unit 2 Review: The Most-Tested Derivative Questions

2. Analyze $g(x) = \int_a^x f(t)\,dt$

4. Net change: $f(b)$ from $f(a)$ and the integral of $f'$

6. Rewrite an integral in terms of $u$