AP Calculus ABAP Exams

AP Calculus AB Unit 8 Review: The Most-Tested Integration Application Questions

Sigma Prep

Math Instructor · 10+ Years Experience

October 24, 2026·14 min read

Key takeaways

Unit weight: 43 of 315 official multiple choice questions (13.7%) and 28.1% of the free-response points, more than any other unit.
Top types: position from velocity, amounts from a rate, displacement and total distance, and average value.
Practice: the calculator questions. 59 of the 83 Unit 8 free-response parts since 2012 were in the calculator part.

Unit 8 carries more of the AP Calculus AB exam than any other unit, mostly through the free response. College Board’s Course and Exam Description (the CED) gives it 10–15% of the multiple choice, and the official exams sit inside that range: 97 of the 719 in-scope multiple choice questions across all 17 official exams from 1969 to 2018 (13.5%), and 43 of 315 on the 7 official practice exams from 2012–2018 (13.7%), 5 to 8 per exam. On the 42-question section that starts in May 2027, that rate is about 6 questions.

The free response leans on it far harder. Unit 8 earned 28.1% of the free-response points on the 2012–2026 exams, more than any other unit, and 42 of the 86 free-response questions since 2012 had a Unit 8 part, on 14 of the 15 exams (all but the shortened 2020 exam), usually 3 per exam. Three kinds of free-response question carry most of it:

The rate question, usually question 1 with a calculator: an amount from a rate (11 questions since 2012), the greatest or least amount when something flows in and out (6), an average value (13 questions with an average-value part), and what an integral means, with units (7).
Particle motion: position from velocity, displacement and total distance, on 13 of the 15 exams.
The region question: the area of a region and a volume built on it, on 10 of the 15 exams, 6 of them in the calculator part.

The 8 most-tested Unit 8 question types

Ranked by share of the whole exam. The eight add up to about 15.4% of the exam, out of 20.5% for all of Unit 8. Two of them, position from velocity and amounts from a rate, are the second and third biggest question types of any unit. Nine smaller types follow.

1. Position or velocity from an initial condition

The position at time $b$ is the known position plus the integral of the velocity from the known time: $x(b) = x(t_0) + \int_{t_0}^{b} v(t)\,dt$ . Velocity from acceleration works the same way.

Topic 8.22.9% of the exam5 of 7 practice exams (6 questions)11 FRQs since 2012

Try it

A particle moves along the

x

-axis with velocity

v(t) = 3t^2 - 4t

, and

x(1) = 5

. Find

x(3)

Solution

$x(3) = 5 + \int_1^3 (3t^2 - 4t)\,dt = 5 + \big[t^3 - 2t^2\big]_1^3 = 5 + 9 - (-1) = 15$ .

Common mistake

leaving out the starting position (10), or integrating from 0 when the position is known at

t = 1

(14).

Practice: Displacement and Distance worksheet · AP Unit 8 worksheet

2. Amount from a rate in context

The amount at time $T$ is the starting amount plus the integral of the rate in (minus the integral of a rate out), in the units of the amount. On the exam this is almost always a calculator question.

Topic 8.32.7% of the exam7 of 7 practice exams (8 questions)11 FRQs since 2012

Try itCalculator

A tank holds 200 gallons of water at

t = 0

. Water flows in at

R(t) = 20\sin\left(\frac{t^2}{8}\right)

gallons per hour. How much water is in the tank at

t = 4

hours?

Solution

$200 + \int_0^4 R(t)\,dt = 200 + 39.905 = 239.905$ gallons.

Common mistake

giving only the integral (39.905 gallons, the water that flowed in), or adding a rate out instead of subtracting it.

Practice: Net Change worksheet · AP Unit 8 worksheet

3. Displacement and total distance

Displacement is the integral of the velocity; total distance is the integral of the speed, $|v(t)|$ . Split the interval where the velocity changes sign.

Topic 8.22.1% of the exam6 of 7 practice exams (6 questions)8 FRQs since 2012

Try it

For

v(t) = t^2 - 4t + 3

0 \le t \le 3

, find the displacement and the total distance.

Solution

$v = (t - 1)(t - 3)$ changes sign at $t = 1$ . The displacement is $\int_0^3 v\,dt = 0$ . The distance is $\int_0^1 v\,dt - \int_1^3 v\,dt = \frac{4}{3} + \frac{4}{3} = \frac{8}{3}$ .

Common mistake

giving the displacement for the distance. Here that answer is 0 although the particle moved

\frac{8}{3}

units: where

v < 0

the motion still adds to the distance.

Practice: Displacement and Distance worksheet · AP Unit 8 worksheet

4. Average value of a function

The average value of $f$ on $[a, b]$ is $\frac{1}{b - a}\int_a^b f(x)\,dx$ . The Mean Value Theorem for integrals adds a $c$ where $f(c)$ equals it, a common follow-up on the free response.

Topic 8.12.0% of the exam3 of 7 practice exams (3 questions)9 FRQs since 2012

Try it

Find the average value of

f(x) = x^2 + 1

[0, 3]

, and the

c

[0, 3]

where

f(c)

equals it.

Solution

$\dfrac{1}{3}\int_0^3 (x^2 + 1)\,dx = \dfrac{1}{3}(12) = 4$ , and $c^2 + 1 = 4$ gives $c = \sqrt{3}$ .

Common mistake

computing the average rate of change,

\frac{f(3) - f(0)}{3} = 3

, or forgetting to divide by the length of the interval (12).

Practice: Average Value of a Function worksheet · AP Unit 8 worksheet

5. Area between curves

Find where the curves meet (on the calculator when needed), then integrate top minus bottom between those $x$ -values.

Topic 8.41.8% of the exam2 of 7 practice exams (2 questions)6 FRQs since 2012

Try it

Find the area of the region between

y = 4 - x^2

and

y = x + 2

Solution

the curves meet where $x^2 + x - 2 = 0$ , at $x = -2$ and $x = 1$ , and the parabola is on top, so the area is $\int_{-2}^{1} (2 - x - x^2)\,dx = \frac{9}{2}$ .

Common mistake

integrating over the wrong interval (from 0 to 1 gives

\frac{7}{6}

) or bottom minus top (

-\frac{9}{2}

). An area is never negative.

Practice: Area Between Two Curves worksheet · AP Unit 8 worksheet

6. Volume with square or rectangle cross sections

Each cross section stands on a chord of the base. A square on a chord of length $s$ has area $s^2$ ; integrate that area across the base.

Topic 8.71.7% of the exam3 of 7 practice exams (3 questions)9 FRQs since 2012

Try it

The region in type 5 is the base of a solid whose cross sections perpendicular to the

x

-axis are squares. Find the volume.

Solution

the side is $s = 2 - x - x^2$ , so the volume is $\int_{-2}^{1} (2 - x - x^2)^2\,dx = \frac{81}{10}$ .

Common mistake

multiplying by

\pi

out of habit from discs (

\frac{81\pi}{10}

), or forgetting to square the side. A square cross section has no

\pi

Practice: Known Cross Sections worksheet · AP Unit 8 worksheet

7. Rate in, rate out: the greatest amount

The amount changes at the rate in minus the rate out. It can be greatest only where that difference is 0 or at an endpoint, so compare the amount at every candidate.

Topic 8.31.2% of the exam1 of 7 practice exams (1 question)6 FRQs since 2012, all in the calculator part

Try it

Water enters a tank at

E(t) = 10 + t

and leaves at

L(t) = 3t

liters per minute, for

0 \le t \le 8

. The tank holds 50 liters at

t = 0

. When does it hold the most, and how much?

Solution

$E - L = 10 - 2t = 0$ at $t = 5$ . The amount is 50 at $t = 0$ , $50 + \int_0^5 (10 - 2t)\,dt = 75$ at $t = 5$ , and 66 at $t = 8$ , so the most is 75 liters at $t = 5$ .

Common mistake

stopping at the critical point without checking the endpoints, or answering with the time when the question asks for the amount (or the other way around). On the free response the comparison of candidates is the justification.

Practice: Net Change worksheet · AP Unit 8 worksheet

8. Volume about a line other than the axes (free response)

Revolving about $y = k$ , each radius is the distance from a curve to that line. With a hole, the volume is $\pi\int (R^2 - r^2)\,dx$ , outer radius squared minus inner radius squared. It is often part (c) of the region question.

Topics 8.10 and 8.121.1% of the examno multiple choice questions5 FRQs since 2012

Try it

The region between

y = x

and

y = x^2

for

0 \le x \le 1

is revolved about the line

y = 2

. Find the volume.

Solution

the parabola is farther from $y = 2$ , so $R = 2 - x^2$ and $r = 2 - x$ : $\pi\int_0^1 \left[(2 - x^2)^2 - (2 - x)^2\right] dx = \frac{8\pi}{15}$ .

Common mistake

measuring the radii from the

x

-axis. That gives the volume about the

x

-axis instead,

\frac{2\pi}{15}

Practice: Volume by Disks and Washers worksheet · AP Unit 8 worksheet

Nine more Unit 8 types, lower priority

9. Disc method about the $x$ - or $y$ -axis

A disc of radius $r$ has area $\pi r^2$ . About the $y$ -axis, write the radius in terms of $y$ and integrate $dy$ .

Topic 8.90.9% of the exam2 of 7 practice exams (2 questions)3 FRQs since 2012

Try it

The region under

y = \sqrt{x}

for

0 \le x \le 4

is revolved about the

x

-axis. Find the volume.

Solution

$\pi\int_0^4 (\sqrt{x})^2\,dx = \pi\int_0^4 x\,dx = 8\pi$ .

Common mistake

leaving out

\pi

(8) or the square (

\frac{16\pi}{3}

Practice: Volume by Disks and Washers worksheet · AP Unit 8 worksheet

10. Area under one curve

Area is positive. Where the graph is below the axis, the integral is negative, so split at the zeros and add absolute values.

Topic 8.40.9% of the exam2 of 7 practice exams (2 questions)2 FRQs since 2012

Try it

Find the area between the

x

-axis and

y = x^2 - 4

for

0 \le x \le 3

Solution

the graph is below the axis on $[0, 2]$ and above it on $[2, 3]$ : $\frac{16}{3} + \frac{7}{3} = \frac{23}{3}$ .

Common mistake

giving the integral,

\int_0^3 (x^2 - 4)\,dx = -3

, as the area.

Practice: Definite Integrals worksheet · AP Unit 8 worksheet

11. Average value from a graph or a given value

An average value $k$ on $[a, b]$ means $\int_a^b f(x)\,dx = k(b - a)$ . From a graph, find the integral from areas, with area below the axis counted as negative.

Topic 8.10.8% of the exam5 of 7 practice exams (6 questions)4 FRQs since 2012

Try it

The average value of

f

[2, 6]

is 5. What is

\int_2^6 f(x)\,dx

Solution

$5 \times (6 - 2) = 20$ .

Common mistake

dividing when you should multiply:

\frac{5}{4}

Practice: Average Value of a Function worksheet · AP Unit 8 worksheet

12. Washer method

A washer has area $\pi R^2 - \pi r^2$ , the outer radius squared minus the inner radius squared.

Topic 8.110.5% of the exam1 of 7 practice exams (1 question)no FRQs since 2012 about an axis (the last was in 2007)

Try it

Revolve the region between

y = x

and

y = x^2

0 \le x \le 1

, about the

x

-axis.

Solution

$R = x$ and $r = x^2$ , so $\pi\int_0^1 (x^2 - x^4)\,dx = \frac{2\pi}{15}$ .

Common mistake

squaring the difference of the radii:

\pi\int_0^1 (x - x^2)^2\,dx = \frac{\pi}{30}

Practice: Volume by Disks and Washers worksheet · AP Unit 8 worksheet

13. Set up an integral in context

The integral of a rate over an interval is the total change over that interval; the integral of a density over a length is a total count.

Topic 8.30.4% of the exam1 of 7 practice exams (1 question)2 FRQs since 2012

Try it

Along a 20-mile road, the density of houses

x

miles from the start is

f(x)

houses per mile. Which expression gives the number of houses on the road?

Solution

$\int_0^{20} f(x)\,dx$ . Houses per mile times miles is houses.

Common mistake

reading the integral of a rate as a rate, or multiplying by the length of the road as well (

20\int_0^{20} f(x)\,dx

Practice: Net Change worksheet · AP Unit 8 worksheet

14. Find k from a given area

The line $x = k$ splits a region in half when the integral on one side equals half the area. That is rarely the midpoint.

Topic 8.40.4% of the examno multiple choice since 19982 FRQs since 2012 (2012, 2014)

Try it

The region under

y = 2x

for

0 \le x \le 4

has area 16. For what

k

does the line

x = k

divide it into two regions of equal area?

Solution

$\int_0^k 2x\,dx = k^2 = 8$ , so $k = 2\sqrt{2} \approx 2.828$ .

Common mistake

splitting the width instead of the area:

k = 2

leaves an area of only 4 on the left.

Practice: Area Between Two Curves worksheet · AP Unit 8 worksheet

15. Area when curves cross more than once

Where the curves switch, start a new integral (or integrate $|f - g|$ ). The net integral can cancel.

Topic 8.60.3% of the exam1 of 7 practice exams (1 question)2 FRQs since 2012 (2015, 2026)

Try it

Find the total area between

y = x^3

and

y = x

for

-1 \le x \le 1

Solution

they cross at $x = -1$ , 0 and 1. $\int_{-1}^{0} (x^3 - x)\,dx + \int_0^1 (x - x^3)\,dx = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ .

Common mistake

integrating straight across the crossing:

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

Practice: Area Between Two Curves worksheet · AP Unit 8 worksheet

16. Volume with triangle or semicircle cross sections

A semicircle with diameter $s$ has area $\frac{\pi}{8}s^2$ ; an equilateral triangle with side $s$ , $\frac{\sqrt{3}}{4}s^2$ ; an isosceles right triangle with a leg $s$ , $\frac{1}{2}s^2$ .

Topic 8.80.3% of the exam1 of 7 practice exams (1 question)1 FRQ since 2012 (2014)

Try it

The region under

y = \sqrt{x}

for

0 \le x \le 4

is the base of a solid whose cross sections perpendicular to the

x

-axis are semicircles with diameters in the base. Find the volume.

Solution

$\int_0^4 \frac{\pi}{8}(\sqrt{x})^2\,dx = \frac{\pi}{8}(8) = \pi$ .

Common mistake

using the chord as a radius,

\frac{\pi}{2}s^2

, which gives

4\pi

, or a whole circle on the diameter (

2\pi

Practice: Known Cross Sections worksheet · AP Unit 8 worksheet

17. Explain what an integral means, with units (free response)

The integral of a rate over $[a, b]$ is the change in the quantity from $t = a$ to $t = b$ , in the units of the quantity.

Topic 8.30.3% of the examno multiple choice questions7 FRQs since 2012

Try it

W(t)

is the temperature of a cup of tea, in °F,

t

minutes after it is poured, and

\int_0^{10} W'(t)\,dt = -12

. What does this mean?

Solution

the tea’s temperature fell by 12°F over the first 10 minutes.

Common mistake

giving the units of the rate (°F per minute), or describing the temperature at

t = 10

instead of the change over the interval.

Practice: Net Change worksheet · AP Unit 8 worksheet

What’s rising and fading in Unit 8

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.

Fading: area between curves on multiple choice (1.49% of recent exams against 3.16% across all years, confirmed by teacher material). It has only 2 of its 12 official multiple choice questions on the 2012–2018 practice exams, though the region question still asks for an area on the free response.
Fading: the washer method about an axis (0.16% against 1.47%, confirmed by teacher material). Its last free-response appearance was in 2007; recent region questions revolve about a line other than an axis instead.
New: curves that cross more than once (0.62% against 0.21%), from one 2018 multiple choice question and the free response in 2015 and 2026.
Steady and big: motion and accumulation. Amounts from a rate are 3.17% of recent exams against 2.33% across all years, and position from velocity is level (3.04% against 3.01%).
On the newest free response, Unit 8 went from 27.5% of the points in 2012–2018 to 28.7% in 2019–2026. Displacement and total distance (1.6% to 3.6%) and average value (2.1% to 3.6%) went up; amounts from a rate (4.8% to 2.8%) and finding k from an area (1.3% to 0%) went down. One question is about 2.3% of a window’s points, so each of those is about one question.

In the CED, rarely asked in multiple choice

Area between curves as functions of y (topic 8.5) and discs and washers about a line other than an axis (topics 8.10 and 8.12) have never been the main topic of an official multiple choice question. The free response does ask for volumes about other lines (type 8 above).
Curves that cross more than once (topic 8.6) has 1 official multiple choice question, and triangle and semicircle cross sections (8.8) and the washer method (8.11) 2 each.
Arc length (topic 8.13) is BC only.

What students over- and under-practice in Unit 8

We tagged 10,263 multiple choice questions from teacher worksheets, review packets and prep books (reposted official questions and non-AB content removed). Unit 8 is 9.60% of them, against 13.49% of the 719 in-scope official multiple choice questions from 1969–2018, so practice gives the unit less than the exam does. Per type:

Question type	Practice vs official share	Ratio
Washer method	0.90% vs 0.28%	3.2× over
Average value of a function	1.52% vs 1.67%	0.9×, about even
Disc method about an axis	0.71% vs 0.83%	0.9×, about even
Area between curves	1.21% vs 1.67%	0.7× under
Square or rectangle cross sections	0.60% vs 0.83%	0.7× under
Amount from a rate in context	0.87% vs 1.53%	0.6× under
Displacement and total distance	0.95% vs 1.67%	0.6× under
Position or velocity from an initial condition	1.05% vs 1.95%	0.5× under
Area under one curve	0.43% vs 0.83%	0.5× under
Average value from a graph or a given value	0.32% vs 0.97%	0.3× under

Official counts per type are small (2 to 14 questions), so read the ratios as directions.

Against the whole exam, motion and accumulation are badly under-practiced. Position from velocity gets 1.05% of practice questions against 2.91% of the exam, amounts from a rate 0.87% against 2.72%, and rate in, rate out only 0.11% against 1.17%, because most of their weight is on the free response.
The washer method is over-practiced while the exam has moved on to volumes about other lines and cross sections, and square cross sections get about a third of their share of the exam (0.60% against 1.66%).
Two free-response types have no multiple choice version: volumes about a line other than an axis (5 free-response questions since 2012) and explaining an integral with units (7). A multiple choice diet never practices either.

Calculator note

Unit 8 is the most calculator-heavy unit. Across all 17 official exams, 51 of its 97 multiple choice questions were in the calculator section (53%; the next highest is Unit 5 at 43%), and on the 2012–2018 practice exams 28 of 43 were. The calculator questions are amounts from rate models, average values, distances, and areas and volumes where the curves meet at points only a calculator finds.

On free response, 59 of the 83 Unit 8 parts since 2012 were in the calculator part. Store the intersection points in the calculator rather than retyping rounded values, and give answers to three decimal places. The other 24, mostly region questions on curves you can integrate by hand (12) and particle motion (7), are done without one.

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 8

Our free AP Calculus AB Unit 8 worksheet has 20 original multiple choice questions on applications of integration, weighted by the exam: 2 each on position or velocity from an initial condition, displacement and total distance, and amounts from a rate, and 1 each on the other fourteen multiple choice types, from average value and area between curves (sliced in x or in y) to cross sections, washers and discs (about the axes or about another line). Part A has 8 questions without a calculator and Part B has 12 with one, as the exam leans on the calculator for this unit; 10 or 11 of the 12 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 8 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 8 Progress Check in AP Classroom.

Common Questions

The CED gives Unit 8 10–15% of the multiple choice, and the official exams match: 43 of 315 multiple choice questions (13.7%) on the 7 official practice exams from 2012–2018, 5 to 8 per exam. The free response gives it far more: 28.1% of the points on the 2012–2026 exams, more than any other unit, with a Unit 8 part on 14 of the 15 exams.

Three kinds. The rate question (an amount from a rate, the greatest amount when something flows in and out, an average value, and what an integral means with units), usually question 1 with a calculator. Particle motion (position from velocity, displacement and total distance), on 13 of the 15 exams from 2012 to 2026. And the region question (an area and a volume on that region), on 10 of the 15.

Displacement is the integral of velocity, the net change in position. Total distance is the integral of speed, |v(t)|: split the interval where the velocity changes sign and add the pieces as positive numbers. A particle that goes out and comes back has displacement 0 but a positive distance.

Often. Across all 17 official exams, 51 of the 97 Unit 8 multiple choice questions were in the calculator section, the most of any unit, and 59 of the 83 Unit 8 free-response parts since 2012 were in the calculator part. Amounts from rate models, distances, and areas and volumes where the curves meet at points only a calculator finds all need one; answers are given to three decimal places.

No. Arc length (topic 8.13) is BC only. AB Unit 8 covers average value, motion with integrals, accumulation in context, area between curves, volumes with known cross sections, and discs and washers about the axes and about other lines.

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AP Calculus AB Unit 8 Review: The Most-Tested Integration Application Questions

The 8 most-tested Unit 8 question types

1. Position or velocity from an initial condition

2. Amount from a rate in context

3. Displacement and total distance

4. Average value of a function

5. Area between curves

6. Volume with square or rectangle cross sections

7. Rate in, rate out: the greatest amount

8. Volume about a line other than the axes (free response)

Nine more Unit 8 types, lower priority

9. Disc method about the xxx- or yyy-axis

10. Area under one curve

11. Average value from a graph or a given value

12. Washer method

13. Set up an integral in context

14. Find k from a given area

15. Area when curves cross more than once

16. Volume with triangle or semicircle cross sections

17. Explain what an integral means, with units (free response)

What’s rising and fading in Unit 8

In the CED, rarely asked in multiple choice

What students over- and under-practice in Unit 8

Calculator note

How we counted

Practice Unit 8

Common Questions

How much of the AP Calculus AB exam is Unit 8?+

What are the most common Unit 8 free-response questions?+

What is the difference between displacement and total distance?+

Do I need a calculator for Unit 8 questions?+

Is arc length on AP Calculus AB?+

Practice this unit: free AP Calculus AB Unit 8 worksheet

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9. Disc method about the $x$ - or $y$ -axis