AP Calculus AB Unit 8 FRQ Practice: Applications of Integration

Q: What is on AP Calculus AB Unit 8?

Unit 8, Applications of Integration, has 12 AB topics in the College Board's course description: the average value of a function on an interval; position, velocity and acceleration with integrals, including displacement and total distance; accumulation functions and definite integrals in applied contexts; the area between curves as functions of x, as functions of y, and for curves that cross more than twice; volumes with square and rectangle cross sections, and with triangle and semicircle cross sections; and volumes of revolution by discs and washers, about the x- or y-axis and about other lines. Topic 8.13 (arc length) is BC only. The course description weights the unit at 10–15% of the multiple-choice section, and a Unit 8 question was on the free-response section of 14 of the 15 exams from 2012 to 2026.

Q: How is an AP Calculus AB free-response question scored?

Each question is worth 9 points, spread over its three or four parts. Points are earned for specific pieces of work: a setup, an antiderivative, a value with its units, a conclusion with its reason. The scoring guide lists every point and what earns it, then a worked solution for each part. On the exam the six free-response questions are half of the score.

Q: Can students use a calculator?

Only where the exam allows one. On this set, questions 1–3 are Part A, where a graphing calculator is required, and question 4 is Part B, without one. A calculator question has parts that need one (a value to three decimal places, an equation solved on the calculator); its other parts can be done by hand, as on the exam.

Q: How were the questions chosen?

We tagged every free-response part from 2012 to 2026 by the skill it tests and the setting of its question. The Unit 8 settings the exam leans on most get the most questions here, each in its own context with its own parts.

Q: Are these real AP exam questions?

No. Every question here is original, written by Sigma Prep in the style of the exam: the settings, functions, numbers and wording are our own, and no official question or scoring guideline is copied or reworded. Every value and statement in the scoring guide was checked by an independent computer-algebra check before the set was published.

Q: Is there a scoring guide?

Yes. Each version has its own: open it under the questions, or download the PDF, which prints the questions with room to write and then the scoring guide on its own pages. There is also a PDF without the scoring guide to hand out.

Four exam-style free-response questions on average value, motion, accumulation, area and volume: three with a graphing calculator, one without. Each is scored out of 9, with a scoring guide and a printable PDF with room to write, free to use in class. Good practice before the Unit 8 Progress Check in AP Classroom. Multiple choice for this unit: Unit 8 worksheet.

Sigma Prep · sigmaprep.io/worksheets/ap-calculus-ab

Name

AP Calculus AB Unit 8 Free Response: Applications of Integration

Date Version A

Directions: Show your work in the space under each part: the setup, the values you use and your reasoning. A justification needs a mathematical reason, with the conditions of any theorem or test you use checked. An answer with no supporting work may earn no credit. Give decimal answers correct to three decimal places unless a part says otherwise. Each question is worth 9 points. Unless a question says otherwise, the domain of a function f is every real number x for which f(x) is a real number.

Part A: A graphing calculator is required.

Questions 1–3

Set your calculator to radian mode. When you use it to solve an equation, find a derivative at a point or evaluate a definite integral, write down the equation, function or integral you used.

1.
Let $f$ and $g$ be the functions $f (x) = 3 \cos x$ and $g (x) = x^{2} - x$ . The shaded region $R$ in the figure above is the region enclosed by their graphs. The graphs of $f$ and $g$ cross at $x = A$ and at $x = B$ , where $A < B$ .
1. (a)
  Determine the area of region $R$ . Show the setup for your calculations.
2. (b)
  Region $R$ is the base of a solid. When the solid is sliced perpendicular to the $x$ -axis, every slice is a rectangle that stands on $R$ and is $2 x + 3$ units tall. What is the volume of the solid? Show the setup for your calculations.
1. (c)
  Revolve $R$ about the horizontal line $y = - 2$ . Write an integral expression for the volume of the solid this produces, but do not evaluate it.
2. (d)
  There is exactly one number $x$ with $0 < x < 1$ at which the graphs of $f$ and $g$ have parallel tangent lines. Find this value of $x$ . Show the setup for your calculations.
2.
After a software update is released, calls come in to a company's help line at a rate modeled by $P (t) = 180 t e^{- t / 2}$ calls per hour, with $t$ in hours after the release and $0 \leq t \leq 6$ .
1. (a)
  Write an integral expression, without evaluating it, for the total number of calls that come in to the help line during the interval $1 \leq t \leq 6$ .
2. (b)
  What is the average value of $P (t)$ from $t = 1$ to $t = 6$ ? Show the setup for your calculations.
1. (c)
  At time $t = 5$ , is the rate at which calls come in increasing or decreasing? Give a reason for your answer.
2. (d)
  The agents on duty can pick up calls at a constant rate of 105 calls per hour; calls that come in faster than that are put on hold. For $a \leq t \leq 4$ , the number of calls on hold is given by $Q (t) = \int_{a}^{t} (P (x) - 105) d x$ , where $a$ is the time at which the line first begins to form. What is the greatest number of calls on hold for $a \leq t \leq 4$ ? Give your answer to the nearest whole number. Justify your answer.
3.
For $0 \leq t \leq 4$ seconds, two drones fly above a road along a straight line that serves as the $x$ -axis. Drone $P$ has velocity $v_{P} (t) = 3 \cos ⁣ (\frac{t^{2}}{3}) - 0.5$ and starts at $x = 4$ when $t = 0$ . Drone $Q$ has velocity $v_{Q} (t) = \sqrt{t + 1} \cos (0.8 t) - 0.5$ and starts at $x = - 5$ when $t = 0$ . Velocities are in meters per second.
1. (a)
  Find $x_{P} (2)$ and $x_{Q} (2)$ , the positions of drone $P$ and drone $Q$ at time $t = 2$ . Show the setup for your calculations.
2. (b)
  At time $t = 2$ , are $P$ and $Q$ moving toward each other, or are they moving away from each other? Explain your reasoning.
1. (c)
  Find the acceleration of drone $Q$ at time $t = 2$ , showing the setup for your calculations. At that time, is the speed of $Q$ increasing or decreasing? Give a reason for your answer.
2. (d)
  What is the total distance that drone $P$ travels from $t = 0$ to $t = 4$ ? Show the setup for your calculations.

Part B: No calculator is allowed.

Question 4

4.
Carts $C$ and $D$ each move on a level air track modeled by the $x$ -axis during the time interval $0 \leq t \leq 4$ . At time $t$ , cart $C$ is at position $x_{C} (t) = t + \frac{4}{t + 1}$ , and cart $D$ has velocity $v_{D} (t) = \frac{- 2 (t - 3)}{t^{2} - 6 t + 10}$ .
1. (a)
  Use the position function of cart $C$ to determine its velocity when $t = 2$ . Show the work that leads to your answer.
2. (b)
  Over $0 < t < 4$ , determine the open time intervals during which $C$ and $D$ head in opposite directions along the axis. Give a reason for your answer.
1. (c)
  Decide whether cart $D$ is gaining speed, losing speed, or neither at the instant $t = 1$ . Give a reason for your answer.
2. (d)
  Cart $D$ starts at $x = 8$ when $t = 0$ . Where on the axis is it when $t = 2$ ? Show the work that leads to your answer.
  End of Version A

Free AP Calculus AB worksheets at sigmaprep.io/worksheets/ap-calculus-abOriginal questions written by Sigma Prep. AP is a registered trademark of the College Board, which is not affiliated with this sheet.

Scoring guide · Version A

Question 1Area, rectangular cross sections, a volume setup and parallel tangents on a region (calculator) · calculator · 9 points

(a) 2 points Area of a region · Topic 8.4

1 point: form of the integrand: $f (x) - g (x)$ (or $g (x) - f (x)$ with the sign resolved) in a definite integral
1 point: answer: $4.726$

Solution

$f (x) = g (x)$ at $x = - 0.930$ and $x = 1.389$ (calculator); on that interval the graph of $f$ is above the graph of $g$ .
Area $= \int_{- 0.930}^{1.389} (f (x) - g (x)) d x = 4.726$ .

(b) 2 points Volume by cross sections · Topic 8.7

1 point: form of the integrand: the product of $2 x + 3$ and $f (x) - g (x)$ in a definite integral
1 point: answer: $16.236$

Solution

At $x$ the base of the rectangle runs across $R$ , with length $f (x) - g (x)$ ; its area is $(2 x + 3) (f (x) - g (x))$ .
Volume $= \int_{- 0.930}^{1.389} (2 x + 3) (f (x) -$ $g (x)) d x =$ $16.236$ .

(c) 3 points Volume of revolution · Topic 8.12

1 point: form of the integrand: $R^{2} - r^{2}$ , each radius a difference between one of the curves and the constant $- 2$
1 point: integrand: $(f (x) + 2)^{2} - (g (x) + 2)^{2}$
1 point: limits $x = - 0.930$ and $x = 1.389$ , the constant $π$ and $d x$ (only with the form point)

Solution

The line $y = - 2$ lies below $R$ : the outer radius is $f (x) + 2$ and the inner radius is $g (x) + 2$ .
Volume $= π \int_{- 0.930}^{1.389} ((f (x) + 2)^{2} - (g (x) + 2)^{2}) d x =$ $π \int_{- 0.930}^{1.389} ({(3 \cos x + 2)}^{2} - {(x^{2} - x + 2)}^{2}) d x$ .

(d) 2 points Horizontal, vertical or given-slope tangent · Topic 2.2

1 point: $f^{'} (x) = g^{'} (x)$ : $- 3 \sin x = 2 x - 1$
1 point: answer: $x = 0.201$

Solution

$f^{'} (x) = - 3 \sin x$ and $g^{'} (x) = 2 x - 1$ . The tangent lines are parallel where $f^{'} (x) = g^{'} (x)$ : $- 3 \sin x = 2 x - 1$ .
Solving on $0 < x < 1$ (calculator): $x = 0.201$ .

Question 2An arrival rate and a line served at a constant rate · calculator · 9 points

(a) 1 point Total change from a rate · Topic 8.3

1 point: definite integral with the correct limits: $\int_{1}^{6} P (t) d t$

Solution

The total that arrives over an interval is the integral of the arrival rate over that interval:
$\int_{1}^{6} P (t) d t = \int_{1}^{6} 180 t e^{- t / 2} d t$ .

(b) 2 points Average value · Topic 8.1

1 point: uses the average value:
$\frac{1}{5} \int_{1}^{6} P (t) d t$
1 point: answer: $102.333$ calls per hour

Solution

The average value of $P$ on $[1, 6]$ is $\frac{1}{5}$ times its integral over the interval.
On the calculator, $\frac{1}{5} \int_{1}^{6} P (t) d t = 102.333$ calls per hour.

(c) 2 points Increasing or decreasing at a time · Topic 4.3

1 point: considers $P^{'} (5)$
1 point: decreasing, with the reason $P^{'} (5) < 0$

Solution

The rate $P (t)$ is increasing where $P^{'} (t) > 0$ and decreasing where $P^{'} (t) < 0$ .
On the calculator, $P^{'} (5) = - 22.163$ , which is negative.

(d) 4 points Absolute extremum (candidates test) · Topic 8.3

1 point: considers $Q^{'} (t) = P (t) - 105 = 0$
1 point: $a = 0.928$ and the other solution of $P (t) = 105$ , $t = 3.688$ (called $b$ below)
1 point: answer: 49 calls
1 point: justification: the candidates test, with $Q (a) = 0$ , $Q (b)$ and $Q (4)$
Also acceptable: A sign argument: $Q^{'} (t) = P (t) - 105 > 0$ for $a < t < b$ and $Q^{'} (t) < 0$ for $b < t < 4$ , so $Q$ increases and then decreases on $[a, 4]$ , and its greatest value is $Q (b)$ .

Solution

$Q^{'} (t) = P (t) - 105$ , which is 0 where $P (t) = 105$ : at $t = 0.928$ , when the line first forms (so $a = 0.928$ ), and at $t = b = 3.688$ (calculator).
The greatest value of $Q$ on $[a, 4]$ is at $t = a$ , at $t = b$ or at $t = 4$ .
Candidates test: $Q (a) = 0$ , $Q (b) = \int_{0.928}^{3.688} (P (x) - 105) d x = 49.058$ , $Q (4) = 47.883$ (calculator).
The greatest number is $49.058$ , which rounds to 49 calls.

Question 3Two particles from velocity models: positions, relative motion, speed and distance · calculator · 9 points

(a) 3 points Position from velocity · Topic 8.2

1 point: a definite integral: $\int_{0}^{2} v_{P} (t) d t$ or $\int_{0}^{2} v_{Q} (t) d t$
1 point: one position, from its starting position: $x_{P} (2) = 8.018$
1 point: the other position: $x_{Q} (2) = - 4.380$

Solution

Each position at $t = 2$ is the starting position plus the integral of the velocity from $t = 0$ to $t = 2$ .
$x_{P} (2) = 4 + \int_{0}^{2} v_{P} (t) d t = 8.018$ (calculator).
$x_{Q} (2) = - 5 + \int_{0}^{2} v_{Q} (t) d t = - 4.380$ (calculator).

(b) 2 points Two particles relative to each other · Topic 4.2

1 point: the signs of both velocities: $v_{P} (2) = 0.206$ and $v_{Q} (2) = - 0.551$
1 point: away from each other, with the explanation from the directions of motion and the positions from part (a)

Solution

On the calculator, $v_{P} (2) = 0.206$ and $v_{Q} (2) = - 0.551$ : $P$ moves to the right and $Q$ to the left.
From part (a), $x_{P} (2) = 8.018$ and $x_{Q} (2) = - 4.380$ , so $Q$ is on the left.
$Q$ is to the left of $P$ at $t = 2$ , and $Q$ is moving to the left while $P$ is moving to the right, so they are moving away from each other.

(c) 2 points Speeding up or slowing down · Topic 4.2

1 point: setup and value: $a_{Q} (2) = v_{Q}^{'} (2) = - 1.393$
1 point: speed increasing, because $v_{Q} (2)$ and $a_{Q} (2)$ have the same sign

Solution

The acceleration of $Q$ is the derivative of its velocity: $a_{Q} (2) = v_{Q}^{'} (2) = - 1.393$ (calculator), which is negative.
$v_{Q} (2) = - 0.551$ is negative; with the same sign, the speed of $Q$ is increasing.

(d) 2 points Total distance · Topic 8.2

1 point: integral: $\int_{0}^{4} ∣ v_{P} (t) ∣ d t$
1 point: answer: $7.962$

Solution

The total distance is the integral of the speed $∣ v_{P} (t) ∣$ over the interval.
On the calculator, $\int_{0}^{4} ∣ v_{P} (t) ∣ d t = 7.962$ .

Question 4Two objects on a line: one by position, one by velocity · no calculator · 9 points

(a) 2 points Velocity from position · Topic 4.2

1 point: uses the derivative: $v_{C} (t) = x_{C}^{'} (t)$
1 point: answer:
$v_{C} (2) = \frac{5}{9}$

Solution

Velocity is the derivative of position:
$v_{C} (t) = x_{C}^{'} (t) = 1 - \frac{4}{{(t + 1)}^{2}}$ .
At $t = 2$ :
$v_{C} (2) = \frac{5}{9}$ .

(b) 3 points Direction of motion · Topic 4.2

1 point: considers where a velocity is zero or changes sign: $v_{C} (t) = 0$ at $t = 1$ , or $v_{D} (t) = 0$ at $t = 3$
1 point: a correct sign analysis of the velocity of one cart on $0 < t < 4$
1 point: answer: $0 < t < 1$ and $3 < t < 4$ , with the reason that $v_{C} (t)$ and $v_{D} (t)$ have opposite signs there (requires both analyses)

Solution

Each cart moves right where its velocity is positive and left where it is negative. $v_{C} (t) < 0$ exactly when ${(t + 1)}^{2} < 4$ , that is, when $t < 1$ , so $v_{C} (t) = 0$ only at $t = 1$ ; cart $C$ moves left on $0 < t < 1$ .
$v_{D} (t) = \frac{- 2 (t - 3)}{t^{2} - 6 t + 10} = 0$ only at $t = 3$ in $0 < t < 4$ ; testing a time in each interval, cart $D$ moves left on $3 < t < 4$ .
The velocities have opposite signs, so the carts move in opposite directions, on $0 < t < 1$ and $3 < t < 4$ .

(c) 1 point Speeding up or slowing down · Topic 4.2

1 point: speed increasing, because $v_{D} (1) = \frac{4}{5}$ and $v_{D}^{'} (1) = \frac{6}{25}$ have the same sign

Solution

$v_{D} (1) = \frac{4}{5}$ , which is positive. The acceleration is $v_{D}^{'} (t) = - \frac{2}{t^{2} - 6 t + 10} -$ $\frac{(- 2 t + 6) (2 t - 6)}{{(t^{2} - 6 t + 10)}^{2}}$ , so $v_{D}^{'} (1) = \frac{6}{25}$ , which is positive.
Speed increases when velocity and acceleration have the same sign and decreases when they have opposite signs, so the speed is increasing.

(d) 3 points Position from velocity · Topic 8.2

1 point: an integral of $v_{D} (t)$ :
$x_{D} (2) = 8 + \int_{0}^{2} v_{D} (t) d t$
1 point: antiderivative:
$- \ln ⁣ (t^{2} - 6 t + 10)$
1 point: answer: $8 - \ln 2 + \ln 10$

Solution

The position at $t = 2$ is the starting position plus the displacement:
$x_{D} (2) = 8 + \int_{0}^{2} v_{D} (t) d t$ .
An antiderivative of $v_{D} (t) = \frac{- 2 (t - 3)}{t^{2} - 6 t + 10}$ (with $u = t^{2} - 6 t + 10$ ) is $- \ln ⁣ (t^{2} - 6 t + 10)$ .
So the displacement is $- \ln 2 - (- ⁣ \ln 10) = \ln 10 - \ln 2$ , and $x_{D} (2) = 8 - \ln 2 + \ln 10$ .

What does the AP Calculus AB free response ask on Unit 8?

More than any other unit: Unit 8 is about 28% of the free-response points since 2012.

The area and volume of a region: about 38% of Unit 8 free-response points; 1 on this set.
A particle moving along a line: about 28% of Unit 8 free-response points; 2 on this set.
An amount from a rate in context: about 21% of Unit 8 free-response points; 1 on this set.
A table of a rate, then a model: about 9% of Unit 8 free-response points; not on this set.

This set has 4 questions in that mix, 3 with a calculator (about 67% of Unit 8 free-response points are in the calculator section), each scored out of 9 with a scoring guide.

Counted from the official AP Calculus AB free-response questions, 2012–2026. No official question is reproduced here.

AP Calculus AB Unit 8 free-response settings by share of exam points

Free-response settings on this set and their share of official AP Calculus AB free-response points, 2012–2026
Setting	Questions on this set	Share of Unit 8 free-response points	Share of all free-response points	Last on the exam
An arrival rate and a line served at a constant rateCalculator	1 of 4	7%	9%	–
Two particles from velocity models: positions, relative motion, speed and distanceCalculator	1 of 4	5%	9%	–
Area, rectangular cross sections, a volume setup and parallel tangents on a region (calculator)Calculator	1 of 4	10%	7%	–
Two objects on a line: one by position, one by velocityNo calculator	1 of 4	6%	8%	–

A part counts toward a setting when its primary skill is in Unit 8, in the same section, and the setting asks that skill (in the part’s own setting where there is one), on the main exams 2012–2026, by its points; a part that fits two settings is shared between them.

Questions about this free-response set

Unit 8, Applications of Integration, has 12 AB topics in the College Board's course description: the average value of a function on an interval; position, velocity and acceleration with integrals, including displacement and total distance; accumulation functions and definite integrals in applied contexts; the area between curves as functions of x, as functions of y, and for curves that cross more than twice; volumes with square and rectangle cross sections, and with triangle and semicircle cross sections; and volumes of revolution by discs and washers, about the x- or y-axis and about other lines. Topic 8.13 (arc length) is BC only. The course description weights the unit at 10–15% of the multiple-choice section, and a Unit 8 question was on the free-response section of 14 of the 15 exams from 2012 to 2026.

Each question is worth 9 points, spread over its three or four parts. Points are earned for specific pieces of work: a setup, an antiderivative, a value with its units, a conclusion with its reason. The scoring guide lists every point and what earns it, then a worked solution for each part. On the exam the six free-response questions are half of the score.

Only where the exam allows one. On this set, questions 1–3 are Part A, where a graphing calculator is required, and question 4 is Part B, without one. A calculator question has parts that need one (a value to three decimal places, an equation solved on the calculator); its other parts can be done by hand, as on the exam.

We tagged every free-response part from 2012 to 2026 by the skill it tests and the setting of its question. The Unit 8 settings the exam leans on most get the most questions here, each in its own context with its own parts.

No. Every question here is original, written by Sigma Prep in the style of the exam: the settings, functions, numbers and wording are our own, and no official question or scoring guideline is copied or reworded. Every value and statement in the scoring guide was checked by an independent computer-algebra check before the set was published.

Yes. Each version has its own: open it under the questions, or download the PDF, which prints the questions with room to write and then the scoring guide on its own pages. There is also a PDF without the scoring guide to hand out.

AP is a registered trademark of the College Board, which was not involved in making these questions and does not endorse them. All questions and scoring guides are original to Sigma Prep. Frozen 2026-10-24.

AP Calculus AB Unit 8 FRQ Practice: Applications of Integration

Part A: A graphing calculator is required.

Part B: No calculator is allowed.

Scoring guide · Version A

Question 1Area, rectangular cross sections, a volume setup and parallel tangents on a region (calculator) · calculator · 9 points

Question 2An arrival rate and a line served at a constant rate · calculator · 9 points

Question 3Two particles from velocity models: positions, relative motion, speed and distance · calculator · 9 points

Question 4Two objects on a line: one by position, one by velocity · no calculator · 9 points

What does the AP Calculus AB free response ask on Unit 8?

AP Calculus AB Unit 8 free-response settings by share of exam points

More AP Calculus AB free response

Questions about this free-response set

What is on AP Calculus AB Unit 8?+

How is an AP Calculus AB free-response question scored?+

Can students use a calculator?+

How were the questions chosen?+

Are these real AP exam questions?+

Is there a scoring guide?+