AP Calculus BC 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

The rate at which water leaks from a tank is modeled by the function L(t), measured in gallons per hour, where t is the number of hours since the leak began. Which of the following expressions represents the total amount of water, in gallons, that leaked from the tank between hour 2 and hour 5?

A)L(5) - L(2)B)∫[2, 5] L(t) dtC)(L(5) - L(2)) / (5 - 2)D)L(5)

All Questions (14)

A) L(5) - L(2)

B) ∫[2, 5] L(t) dt

C) (L(5) - L(2)) / (5 - 2)

D) L(5)

Correct Answer: B

The definite integral of a rate of change (L(t), in gallons per hour) over an interval gives the net change or total accumulation of that quantity over the interval. Therefore, integrating L(t) from t=2 to t=5 gives the total gallons leaked during that time. [cite: 2889, 2890]

A particle moves along the x-axis with a velocity given by v(t) for t ≥ 0. Which of the following definite integrals gives the net change in the particle's position, also known as displacement, from time t=a to time t=b?

A) v(b)

B) v(b) - v(a)

C) ∫[a, b] v(t) dt

D) ∫[a, b] |v(t)| dt

Correct Answer: C

Velocity is the rate of change of position. The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval. Therefore, the integral of velocity v(t) from a to b gives the net change in position. [cite: 2890]

The rate of consumption of a natural resource is given by R(t), where t is measured in years since 2000 and R(t) is measured in millions of tons per year. The function A(x) = ∫[0, x] R(t) dt is defined. What is the best interpretation of A(10)?

A) The rate of consumption in the year 2010.

B) The average rate of consumption from 2000 to 2010.

C) The total amount of the resource, in millions of tons, consumed from the beginning of 2000 to the beginning of 2010.

D) The year in which consumption reached 10 million tons.

Correct Answer: C

A function defined as an integral, such as A(x), represents an accumulation of the rate of change, R(t). Therefore, A(10) represents the total accumulation of the resource consumed from t=0 (year 2000) to t=10 (year 2010). [cite: 2889]

The temperature of a substance is changing at a rate of T'(t) = 2t - 8 degrees Celsius per minute. What is the net change in the temperature of the substance from t=1 minute to t=4 minutes?

A) -3 degrees Celsius

B) 3 degrees Celsius

C) 9 degrees Celsius

D) -9 degrees Celsius

Correct Answer: D

The net change in temperature is determined by the definite integral of its rate of change over the given time interval. We calculate ∫[1, 4] (2t - 8) dt = [t^2 - 8t] evaluated from 1 to 4. This results in (4^2 - 8*4) - (1^2 - 8*1) = (16 - 32) - (1 - 8) = -16 - (-7) = -9 degrees Celsius. [cite: 2890]

At an amusement park, people enter a ride at a rate modeled by E(t) and leave the ride at a rate modeled by L(t), both measured in people per hour. At t=0, there are 50 people on the ride. Which expression represents the number of people on the ride at t=3 hours?

A) ∫[0, 3] (E(t) - L(t)) dt

B) 50 + ∫[0, 3] (E(t) - L(t)) dt

C) 50 + E(3) - L(3)

D) E(3) - L(3)

Correct Answer: B

The total number of people at a given time is the initial number plus the net change. The net rate of change is E(t) - L(t). The net change from t=0 to t=3 is the integral of this rate, ∫[0, 3] (E(t) - L(t)) dt. Adding the initial amount of 50 gives the total number of people at t=3. [cite: 2891, 2890]

If R'(t) represents the rate of change of a quantity R, then the definite integral of R'(t) over the interval [a, b] gives the:

A) Average value of R on [a, b].

B) Instantaneous rate of change of R at t=b.

C) Average rate of change of R on [a, b].

D) Net change of R from t=a to t=b.

Correct Answer: D

This is the fundamental definition of how a definite integral is used in accumulation problems. The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval, which is equivalent to R(b) - R(a). [cite: 2890]

The population of a colony of insects is growing at a rate of g(t) insects per day. If t is measured in days, what are the units of the quantity represented by ∫[5, 10] g(t) dt?

A) Insects

B) Days

C) Insects per day

D) Insects per day per day

Correct Answer: A

The definite integral accumulates the rate over the interval. The rate g(t) has units of 'insects per day' and the differential dt has units of 'days'. The product inside the integral, g(t)dt, has units of (insects/day) * (day) = insects. Therefore, the value of the integral represents a quantity of insects. [cite: 2889]

A factory's rate of production of cars is given by c(t) cars per week. Let C(x) = ∫[0, x] c(t) dt represent the total number of cars produced in the first x weeks. What is the meaning of C(12) - C(5) = 450?

A) The rate of production at week 7 was 450 cars per week.

B) The factory produced 450 cars during the 7th week.

C) The factory produced a total of 450 cars between the end of week 5 and the end of week 12.

D) The rate of production increased by 450 cars per week between week 5 and week 12.

Correct Answer: C

C(x) is an accumulation function representing the total cars produced by week x. C(12) is the total produced by week 12, and C(5) is the total produced by week 5. Their difference, C(12) - C(5), represents the net change, or the number of cars produced, during the interval from t=5 to t=12. [cite: 2889, 2890]

The altitude of a hot air balloon is changing at a rate of r(t) feet per minute. If the balloon is at 500 feet at t=10 minutes and ∫[10, 30] r(t) dt = -150, what is the altitude of the balloon at t=30 minutes?

A) 150 feet

B) 350 feet

C) 500 feet

D) 650 feet

Correct Answer: B

The definite integral ∫[10, 30] r(t) dt represents the net change in altitude from t=10 to t=30. If A(t) is the altitude at time t, then A(30) - A(10) = ∫[10, 30] r(t) dt. We have A(30) - 500 = -150. Solving for A(30) gives A(30) = 500 - 150 = 350 feet. [cite: 2890]

Grain is being added to a silo at a rate of A(t) cubic feet per hour and removed from the silo at a rate of R(t) cubic feet per hour. At t=4 hours, the silo contains 2000 cubic feet of grain. Which of the following expressions represents the volume of grain in the silo at t=10 hours?

A) ∫[4, 10] (A(t) - R(t)) dt

B) 2000 + ∫[0, 10] (A(t) - R(t)) dt

C) 2000 + ∫[4, 10] (A(t) - R(t)) dt

D) 2000 + A(10) - R(10)

Correct Answer: C

The final amount is the initial amount plus the net change over the interval. The initial amount is 2000 cubic feet at t=4. The net rate of change of the volume is A(t) - R(t). The accumulated net change from t=4 to t=10 is given by the definite integral ∫[4, 10] (A(t) - R(t)) dt. Therefore, the total volume at t=10 is the initial volume plus this accumulated change. [cite: 2891, 2890]

A company's profit is changing at a rate of P'(t) dollars per month. The expression ∫[3, 6] P'(t) dt represents:

A) The company's total profit at the end of the 6th month.

B) The average rate of profit change from month 3 to month 6.

C) The net change in the company's profit from the end of month 3 to the end of month 6.

D) The rate of change of profit at the end of the 6th month.

Correct Answer: C

The definite integral of a rate of change (P'(t)) over an interval ([3, 6]) gives the net change of the original quantity (profit) over that same interval. [cite: 2890]

The number of people in a stadium at time t hours is given by the function P(t). The rate at which people enter is E(t) and the rate at which they leave is L(t). At t=1, there are 10,000 people. The function N(t) is defined as N(t) = 10000 + ∫[1, t] (E(x) - L(x)) dx. Which statement is true?

A) N(t) is the net change in the number of people in the stadium from t=1 to time t.

B) N'(t) represents the total number of people in the stadium at time t.

C) N(t) represents the total number of people in the stadium at time t for t ≥ 1.

D) N(t) is the average number of people in the stadium from t=1 to time t.

Correct Answer: C

The function N(t) is constructed as an initial value (10,000 people at t=1) plus the accumulated net change from that initial time. The integral ∫[1, t] (E(x) - L(x)) dx represents the net change in the number of people from hour 1 to hour t. Therefore, N(t) represents the total number of people in the stadium at any time t ≥ 1. [cite: 2889, 2890, 2891]

The radius of a spherical balloon is increasing at a rate of r'(t) centimeters per second. Which integral represents the net increase in the balloon's radius from t=0 to t=5 seconds?

A) r'(5)

B) r'(5) - r'(0)

C) ∫[0, 5] r'(t) dt

D) (1/5) ∫[0, 5] r'(t) dt

Correct Answer: C

To find the net change (or net increase) of a quantity over an interval, one must integrate the rate of change of that quantity over the interval. The rate of change of the radius is r'(t), and the interval is from t=0 to t=5. [cite: 2890, 2891]

Let F(t) be the amount of fuel in a tank at time t. The fuel is consumed at a rate of R(t) gallons per hour. If F(3) = 50, which expression represents the amount of fuel in the tank at t=0, assuming t=0 is before t=3?

A) 50 - ∫[0, 3] R(t) dt

B) 50 + ∫[0, 3] R(t) dt

C) ∫[0, 3] R(t) dt - 50

D) 50 + R(3) - R(0)

Correct Answer: B

The net change equation is F(3) - F(0) = ∫[0, 3] F'(t) dt. Since fuel is being consumed, the rate of change of the amount of fuel is F'(t) = -R(t). So, F(3) - F(0) = ∫[0, 3] -R(t) dt = -∫[0, 3] R(t) dt. We have 50 - F(0) = -∫[0, 3] R(t) dt. Solving for F(0), we get F(0) = 50 + ∫[0, 3] R(t) dt. [cite: 2890]