AP Calculus AB Practice Quiz: Using Accumulation Functions and Definite Integrals in Applied Contexts

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 14 questions to check your progress.

Question 1 of 14

Water is pumped into a tank at a rate modeled by W(t) gallons per hour, where t is measured in hours. Which of the following expressions represents the total amount of water, in gallons, pumped into the tank during the first 8 hours?

A)W(8)B)W(8) - W(0)C)∫[0, 8] W(t) dtD) (1/8) ∫[0, 8] W(t) dt

All Questions (14)

A) W(8)

B) W(8) - W(0)

C) ∫[0, 8] W(t) dt

D) (1/8) ∫[0, 8] W(t) dt

Correct Answer: C

The definite integral of a rate of change gives the net accumulation. Since W(t) is the rate at which water is pumped in (gallons per hour), integrating this rate from t=0 to t=8 gives the total accumulated volume of water in gallons over that time interval. [cite: 2889, 2891]

The population of a colony of insects is changing at a rate of P'(t) insects per day. If the population at t=10 days is 500, which expression gives the population of the colony at t=30 days?

A) ∫[10, 30] P'(t) dt

B) 500 + ∫[10, 30] P'(t) dt

C) P'(30)

D) 500 + P'(30)

Correct Answer: B

The total amount of a quantity at a certain time is the initial amount plus the net change. The net change in population from t=10 to t=30 is given by the definite integral ∫[10, 30] P'(t) dt. To find the total population at t=30, this net change must be added to the known population at t=10, which is 500. [cite: 2890]

A particle moves along the x-axis with a velocity v(t), measured in meters per second. What is the meaning of the expression ∫[2, 5] v(t) dt?

A) The average velocity of the particle from t=2 to t=5.

B) The total distance traveled by the particle from t=2 to t=5.

C) The acceleration of the particle at t=5.

D) The net change in position (displacement) of the particle from t=2 to t=5.

Correct Answer: D

Velocity is the rate of change of position. The definite integral of a rate of change (velocity) over an interval gives the net change of the quantity (position) over that interval. This net change in position is also known as displacement. [cite: 2890, 2891]

The rate at which a company's profit is changing is given by R(t) dollars per month. What does the expression R(b) - R(a) represent?

A) The total profit earned by the company between month a and month b.

B) The average rate of profit change between month a and month b.

C) The net change in the rate of profit between month a and month b.

D) The net change in profit between month a and month b.

Correct Answer: C

The function R(t) itself is the rate of change of profit. Therefore, R(b) - R(a) represents the net change in this rate, not the net change in the actual profit. The net change in profit would be represented by the integral of R(t) from a to b. [cite: 2890]

A function F(x) is defined by F(x) = ∫[0, x] f(t) dt, where f(t) represents the rate of growth of a plant in centimeters per week. What does F(10) represent?

A) The rate of growth of the plant at week 10.

B) The average growth rate of the plant over the first 10 weeks.

C) The total accumulated growth of the plant during the first 10 weeks.

D) The change in the growth rate from week 0 to week 10.

Correct Answer: C

A function defined as an integral represents an accumulation of a rate of change. Since f(t) is the rate of growth, the integral from 0 to 10, F(10), represents the total accumulated growth (net change in height) from t=0 to t=10. [cite: 2889]

The temperature of a chemical reaction is changing at a rate of T'(t) degrees Celsius per second. If ∫[0, 60] T'(t) dt = -15, what is the correct interpretation of this statement?

A) The temperature at t=60 seconds is -15 degrees Celsius.

B) The temperature decreased by 15 degrees Celsius during the first 60 seconds.

C) The rate of temperature change at t=60 seconds is -15 degrees Celsius per second.

D) The average temperature during the first 60 seconds was -15 degrees Celsius.

Correct Answer: B

The definite integral of the rate of change of a quantity, T'(t), gives the net change of that quantity, T(t). The value of the integral, -15, represents the net change in temperature from t=0 to t=60. A negative value indicates a decrease. [cite: 2890]

The principle that the definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over the interval is an application of:

A) The Mean Value Theorem

B) The Squeeze Theorem

C) The Fundamental Theorem of Calculus

D) The Intermediate Value Theorem

Correct Answer: C

The Fundamental Theorem of Calculus, Part 2, states that if F'(x) = f(x), then ∫[a, b] f(x) dx = F(b) - F(a). In applied contexts, f(x) is the rate of change and F(b) - F(a) is the net change. [cite: 2890]

Oil is leaking from a tanker at a rate of L(t) liters per minute. If t is measured in minutes, what are the units of ∫[0, 60] L(t) dt?

A) liters

B) liters per minute

C) liters per minute squared

D) minutes

Correct Answer: A

The integral of a rate function accumulates the quantity. The units of the definite integral are the product of the units of the function (liters/minute) and the units of the variable of integration (minutes). (liters/minute) * (minutes) = liters. [cite: 2889, 2891]

People enter a concert venue at a rate of E(t) people per hour and leave at a rate of L(t) people per hour. There are 200 people in the venue at time t=1. Which expression represents the number of people in the venue at t=4?

A) ∫[1, 4] (E(t) - L(t)) dt

B) 200 + ∫[1, 4] E(t) dt

C) 200 + ∫[1, 4] (E(t) - L(t)) dt

D) 200 + E(4) - L(4)

Correct Answer: C

The net rate of change of the number of people in the venue is E(t) - L(t). To find the total number of people at t=4, we need the initial number at t=1 (200) plus the accumulated net change from t=1 to t=4. The accumulated net change is the integral of the net rate of change, ∫[1, 4] (E(t) - L(t)) dt. [cite: 2890, 2891]

The amount of a certain medication in a patient's bloodstream at time t is given by A(t). The function A'(t) represents the rate at which the medication level is changing. Which of the following represents the net change in the amount of medication in the bloodstream from t=2 hours to t=6 hours?

A) A(6)

B) A'(6) - A'(2)

C) A(6) - A(2)

D) (A(6) - A(2)) / (6 - 2)

Correct Answer: C

The net change of a quantity A(t) over an interval [a, b] is given by A(b) - A(a). Based on the Fundamental Theorem of Calculus, this is also equal to ∫[a, b] A'(t) dt. Option C directly states the definition of net change for the quantity A(t). [cite: 2890]

A car's velocity is given by v(t) in miles per hour. The position of the car at time t is given by the function s(t) = 10 + ∫[0, t] v(x) dx. What is the physical meaning of the number 10 in this expression?

A) The car's initial velocity.

B) The car's position at time t=0.

C) The car's average velocity over the interval [0, t].

D) The car's net displacement from t=0.

Correct Answer: B

The expression represents the final position as the initial position plus the net change (displacement). The integral ∫[0, t] v(x) dx represents the displacement from t=0 to time t. Therefore, the constant 10 added to this displacement must be the initial position at t=0. We can see this by evaluating s(0) = 10 + ∫[0, 0] v(x) dx = 10 + 0 = 10. [cite: 2889, 2890]

A factory produces phones at a rate of P(t) phones per week. Which of the following definite integrals represents the number of phones produced during the third week?

A) ∫[2, 3] P(t) dt

B) ∫[3, 4] P(t) dt

C) P(3)

D) ∫[0, 3] P(t) dt

Correct Answer: A

The first week is the interval [0, 1], the second week is [1, 2], and the third week is the interval from t=2 to t=3. To find the total accumulation during this specific interval, we must integrate the rate function P(t) over that interval, which is from 2 to 3. [cite: 2891]

The definite integral of the rate of change of a quantity over an interval gives the:

A) average value of the quantity over the interval.

B) instantaneous rate of change at the end of the interval.

C) net change of the quantity over the interval.

D) total amount of the quantity at the start of the interval.

Correct Answer: C

This is the fundamental concept of using integrals in applied contexts. The integral of a rate function (e.g., f'(x)) over an interval [a, b] yields the total accumulated change, or net change, of the original function (f(b) - f(a)) over that interval. [cite: 2890]

The number of bacteria in a petri dish is changing at a rate of B'(t) bacteria per hour. If B(t) represents the number of bacteria at time t, what does B(4) - B(1) represent?

A) The rate of bacterial growth at t=4.

B) The average number of bacteria between t=1 and t=4.

C) The net change in the number of bacteria from t=1 to t=4.

D) The total number of bacteria in the dish at t=4.

Correct Answer: C

The expression B(4) - B(1) represents the value of the quantity at the end of an interval minus the value at the beginning. This is the definition of net change. This value is also equivalent to the definite integral ∫[1, 4] B'(t) dt. [cite: 2890]