AP®MathsCollege BoardCalculus ABExam QuestionsUnit 8: Applications of IntegrationDefinite Integrals in Context

Definite Integrals in Context (College Board AP® Calculus AB): Exam Questions

2 hours48 questions

Select a download format for Definite Integrals in Context

Scroll for more

Select an answer set to view for
Definite Integrals in Context

Easy
Medium
Hard
1
Sme Calculator
2 marks

The rate of flow of a liquid, in liters per minute, can be modeled by f open parentheses t close parentheses equals t over 200 cos open parentheses open parentheses t over 80 close parentheses squared close parentheses for 0 less or equal than t less or equal than 100. Using this model, find the average rate of flow of the liquid over the time interval 0 less or equal than t less or equal than 100. Show the setup for your calculations.

How did you do?

2
Sme Calculator
3 marks

Particle P moves along the x-axis such that, for time t greater than 0, its velocity is given by v subscript P open parentheses t close parentheses equals 3 over t squared. At time t equals 1, the position of particle P is x subscript P open parentheses 1 close parentheses equals 9.

Find x subscript P open parentheses t close parentheses, the position of particle P at time t.

How did you do?

3
Sme Calculator
3 marks

A particle, A, is moving along the x-axis. The velocity of the particle is given by v subscript A open parentheses t close parentheses equals cos open parentheses t to the power of 0.7 end exponent close parentheses for 0 less or equal than t less or equal than 2 pi. At time t equals 0, particle A is at position x equals 3.

A second particle, B, also moves along the x-axis. The velocity of particle B is given by v subscript B open parentheses t close parentheses equals e to the power of 0.2 t end exponent open parentheses 0.6 t minus 3 close parentheses for 0 less or equal than t less or equal than 2 pi. At time t equals 0, particle B is at position x equals 8.

Find the position of particles A and B at time t equals 4.

How did you do?

4a
Sme Calculator
2 marks

The electricity consumption rate of a factory is given by the function C open parentheses t close parentheses equals 50 plus 30 space sin space open parentheses pi over 12 t close parentheses .

Electricity is produced by renewable energy sources at a rate P given by P open parentheses t close parentheses equals 40 plus 5 t.

Electricity consumption and production rates are measured in kilowatts per hour and t is measured in hours since midnight, t equals 0.

How much total electricity is consumed by the factory over the working day from 9 space straight A. straight M. to 5 space straight P. straight M.? Give your answer to the nearest kilowatt hour.

How did you do?

4b
Sme Calculator
2 marks

What is the average rate of renewable electricity production per hour over the working day from 9 space straight A. straight M. to 5 space straight P. straight M.?

How did you do?

5
Sme Calculator
3 marks

A car is driven along a straight road. For 0 less or equal than t less or equal than 3, the car's velocity is given by a differentiable function v left parenthesis t right parenthesis equals 20 plus 5 t minus 0.1 t squared , where t is measured in seconds and v left parenthesis t right parenthesis is measured in meters per second.

Using correct units, explain the meaning of the definite integral 1 third integral subscript 0 superscript 3 20 plus 5 t minus 0.1 t squared space d t in the context of the problem and calculate its value.

How did you do?

6
Sme Calculator
2 marks

When a certain grocery store opens, it has 50 pounds of bananas on a display table. Customers remove bananas from the display table at a rate modeled by f(t) = 10 + (0.8t) \text{sin} \; (t^3/100) for 0 < t \leq 12, where f(t) is measured in pounds per hour and t is the number of hours after the store opened. After the store has been open for three hours, store employees add bananas to the display table at a rate modeled by g(t) = 3 + 2.4 \text{ln} \; (t^2 + 2t) for 3 < t \leq 12, where g(t) is measured in pounds per hour and t is the number of hours after the store opened.

How many pounds of bananas are removed from the display table during the first 2 hours the store is open?

How did you do?

1a
Sme Calculator
2 marks

A child is running along a straight track in a schoolyard. The child's velocity is given by v open parentheses t close parentheses equals 10 e to the power of negative 0.05 t end exponent sin open parentheses pi over 48 t close parentheses for 0 less or equal than t less or equal than 96, where v open parentheses t close parentheses is measured in meters per second, and t is measured in seconds.

Find the distance between the child's position at time t equals 10 seconds and their position at time t equals 70 seconds. Show the setup for your calculations.

How did you do?

1b
Sme Calculator
2 marks

Find the total distance the child runs over the time interval 0 less or equal than t less or equal than 96 seconds. Show the setup for your calculations.

How did you do?

2a
Sme Calculator
3 marks

A particle, P, moves along the x-axis so that its velocity , over the interval 0 less or equal than t less or equal than 12, is given by the differentiable function v subscript P open parentheses t close parentheses equals 18 space cube root of t space sin space open parentheses 0.02 t squared close parentheses, where v subscript P open parentheses t close parentheses is measured in meters per second and t is measured in seconds.

Find the time interval during which the velocity of particle P is at least 20 meters per second. Find the distance traveled by the particle P during the time interval when the velocity of particle P is at least 20 meters per second.

How did you do?

2b
Sme Calculator
3 marks

At time t equals 0, particle P is at position x equals negative 6. A second particle Q, also moves along the x-axis such that x subscript Q open parentheses 7 close parentheses equals 70.

Using the function v subscript P from part (a), approximate the distance between the particles P and Q at time t equals 7.

How did you do?

3
Sme Calculator
3 marks

The density of pollen in a circular meadow ,at a distance r meters from the center of the meadow, is given by an increasing, differentiable function. The pollen density is modeled by the function g open parentheses r close parentheses equals 5 plus 10 e to the power of negative 0.5 r end exponent for 2 less or equal than r less or equal than 8, where g left parenthesis r right parenthesis is measured in micrograms per square meter.

For what value of k, 2 less than k less than 8, is g left parenthesis k right parenthesis equal to the average value of g left parenthesis r right parenthesis on the interval 2 less or equal than r less or equal than 8?

How did you do?

4a
Sme Calculator
2 marks

A particular college has a stall at a high school college fair. The college decides to give out branded pens as advertising. Students take the pens from the stall table at a rate modeled by

f open parentheses t close parentheses equals 6 plus t space sin space open parentheses fraction numerator 3.5 open parentheses t plus 2 close parentheses squared over denominator 5 end fraction close parentheses for 0 less than t less or equal than 5

where f open parentheses t close parentheses is measured in pens per hour and t is the number of hours after the start of the college fair. There are initially 30 pens on the stall table.

After the fair has been running for two hours, the college representatives add more pens to the stall table at a rate modeled by

g open parentheses t close parentheses equals 3 space ln space open parentheses 2 t squared minus t close parentheses for 2 less than t less or equal than 5

How many pens are taken by students in the first 2 hours of the college fair?

How did you do?

4b
Sme Calculator
2 marks

How many pens are on the stall table at time t equals 4?

How did you do?

5
Sme Calculator
4 marks

The velocity of a particle P at time t is given by v open parentheses t close parentheses equals t squared minus 14 t plus 45 on the interval 0 less or equal than t less or equal than 12. Particle P is at position negative 2 at time t equals 0.

Find the position of particle P the first time it changes direction.

How did you do?

6
Sme Calculator
3 marks

The density of bacteria in the petri dish, for 1 \leq r \leq 4, is modelled by the function g defined by g(r) = 2 - 16 (\text{cos}(1.57 \sqrt{r}))^{3}. For what value of k, 1 < k < 4, is g(k) equal to the average value of g(r) on the interval 1 \leq r \leq 4?

How did you do?

7a
Sme Calculator
1 mark

From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by A left parenthesis t right parenthesis equals 450 square root of sin left parenthesis 0.62 t right parenthesis end root, where t is the number of hours after 5 A.M. and A(t) is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.

Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. (t = 1) to 10 A.M. (t = 5).

How did you do?

7b
Sme Calculator
2 marks

Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. (t = 1) to 10 A.M. (t = 5).

How did you do?

8
Sme Calculator
2 marks

An invasive species of plant appears in a fruit grove at time t = 0 and begins to spread. The function C defined by C \left(t\right) = 7.6 \, \text{arctan} \; \left(0.2 t\right) models the number of acres in the fruit grove affected by the species t weeks after the species appears. It can be shown that C ' \left(t\right) = \frac{38}{25 + t^{2}}.

(Note: Your calculator should be in radian mode.)

Find the average number of acres affected by the invasive species from time t = 0 to time t = 4 weeks. Show the setup for your calculations.

How did you do?

9
Sme Calculator
3 marks

A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function f, where f(t) is measured in gallons per second and t is measured in seconds since pumping began. Selected values of f(t) are given in the table.

t (seconds)

0

60

90

120

135

150

f(t) (gallons per second)

0

0.1

0.15

0.1

0.05

0

Using correct units, interpret the meaning of \int_{60}^{135} f(t) \text{d} t in the context of the problem. Use a right Riemann sum with the three subintervals [60, 90], [90, 120], and [120, 135] to approximate the value of \int_{60}^{135} f(t) \text{d} t.

How did you do?

10
Sme Calculator
3 marks

A particle, P, is moving along the x-axis. The velocity of particle P at time t is given by v_{P}(t) = \text{sin}(t^{1.5}) for 0 \leq t \leq \pi. At time t = 0, particle P is at position x = 5. A second particle, Q, also moves along the x-axis. The velocity of particle Q at time t is given by v_{Q}(t) = (t - 1.8) \cdot 1.25^{t} for 0 \leq t \leq \pi. At time t = 0, particle Q is at position x = 10.

Find the positions of particles P and Q at time t = 1.

How did you do?

11
Sme Calculator
3 marks

A teacher also starts reading at time t = 0 minutes and continues reading for the next 10 minutes. The rate at which the teacher reads is modeled by the function W defined by W \left(t\right) = - \frac{3}{10} t^{2} + 8 t + 100, where W \left(t\right) is measured in words per minute. Based on the model, how many words has the teacher read by the end of the 10 minutes? Show the work that leads to your answer.

How did you do?

12
Sme Calculator
2 marks

The rate of flow of gasoline, in gallons per second, can also be modeled by g(t) = \left(\frac{t}{500}\right) \text{cos} \left(\left(\frac{t}{120}\right)^2\right) for 0 \leq t \leq 150. Using this model, find the average rate of flow of gasoline over the time interval 0 \leq t \leq 150. Show the setup for your calculations.

How did you do?

13
Sme Calculator
2 marks

The area, in square feet, of the horizontal cross section at height h feet is modeled by the function f given by f(h) = \dfrac{50.3}{e^{0.2h} + h}.

Based on this model, find the volume of the tank. Indicate units of measure.

How did you do?

14a
Sme Calculator
2 marks

Fish enter a lake at a rate modeled by the function E given by E(t) = 20 + 15 \text{sin} \; \left(\frac{\pi t}{6}\right). Fish leave the lake at a rate modeled by the function L given by L(t) = 4 + 2^{0.1t^2}. Both E(t) and L(t) are measured in fish per hour, and t is measured in hours since midnight (t = 0).

(Note: Your calculator should be in radian mode.)

How many fish enter the lake over the 5-hour period from midnight (t = 0) to 5 A.M. (t = 5)? Give your answer to the nearest whole number.

How did you do?

14b
Sme Calculator
2 marks

What is the average number of fish that leave the lake per hour over the 5-hour period from midnight (t = 0) to 5 A.M. (t = 5)?

How did you do?

15
Sme Calculator
3 marks

A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height h feet is given by the function A, where A(h) is measured in square feet. The function A is continuous and decreases as h increases. Selected values for A(h) are given in the table below.

h (feet)

0

2

5

10

A(h) (sq ft)

50.3

14.4

6.5

2.9

Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the volume of the tank. Indicate units of measure.

How did you do?

16
Sme Calculator
3 marks
Diagram of an inverted cone with a smaller circular water surface inside, labelled radius r and water depth h measured vertically from the surface to the tip

The inside of a funnel of height 10 inches has circular cross sections, as shown in the figure above. At height h, the radius of the funnel is given by

r = \frac{1}{20}(3 + h^{2})

where 0 \leq h \leq 10. The units of r and h are inches.

Find the average value of the radius of the funnel.

How did you do?

1a
Sme Calculator
1 mark

A sports game in a stadium ends at 5 space straight P. straight M. and the rate at which people exit the stadium between 5 space straight P. straight M. and 6 space straight P. straight M. is given by R open parentheses t close parentheses equals 350 square root of sin open parentheses 0.052 t close parentheses end root, where t is the number of minutes after 5 space straight P. straight M. and R open parentheses t close parentheses is measured in people per minute.

Write, but do not evaluate, an integral expression that gives the total number of people that exit the stadium from 5 colon 15 space straight P. straight M. open parentheses t equals 15 close parenthesesto 5 colon 45 space straight P. straight M. open parentheses t equals 45 close parentheses.

How did you do?

1b
Sme Calculator
2 marks

Find the average value of the rate, in people per minute, at which people exit the stadium from 5 colon 15 space straight P. straight M. open parentheses t equals 15 close parenthesesto 5 colon 45 space straight P. straight M. open parentheses t equals 45 close parentheses.

How did you do?

1c
Sme Calculator
4 marks

A line to exit the stadium begins to form as soon as R open parentheses t close parentheses reaches 300. The number of people in line at time t, for a less or equal than t less or equal than 50, is given by Q open parentheses t close parentheses equals integral subscript a superscript t open parentheses R open parentheses x close parentheses minus 300 close parentheses space italic d x, where a is the time when a line first begins to form. To the nearest whole number, find the greatest number of people in line to exit the stadium in the time interval a less or equal than t less or equal than 50. Justify your answer.

How did you do?

2
Sme Calculator
4 marks
A graph with two connected shapes: an semicircle from (-5, 0) to (-1, 0) and a line from (0, -2) to (1, 2), and a line from (1,2) to (5, -2).
Graph of f'

The function f is defined on the closed interval [-5, 5]. The graph of f to the power of apostrophe, the derivative of f, consists of two line segments and a semicircle, as shown in the figure. It is known that f open parentheses 4 close parentheses equals 2.

Find f open parentheses 0 close parentheses and f open parentheses negative 5 close parentheses.

How did you do?

3a
Sme Calculator
2 marks

For 0 less or equal than t less or equal than 16, a particle moves along the x-axis. The velocity of the particle at time t is given by v open parentheses t close parentheses equals sin space open parentheses pi over 8 t close parentheses.

For 0 less or equal than t less or equal than 16, when is the particle moving to the right?

How did you do?

3b
Sme Calculator
3 marks

Find the total distance traveled by the particle from time t equals 0 to time t equals 12.

How did you do?

3c
Sme Calculator
3 marks

The particle is at position x equals negative 8 over pi at time t equals 0. Find the position of the particle at time t equals 2.

How did you do?

4a
Sme Calculator
2 marks

Water flows into a fountain at a rate modeled by the function r given by

r open parentheses t close parentheses equals open curly brackets table row cell 15 open parentheses 1 minus e to the power of negative 0.1 t end exponent close parentheses end cell cell for space 0 less or equal than t less or equal than 12 end cell row 0 cell for space t greater than 12 end cell end table close

where r open parentheses t close parentheses is measured in liters per minute and t is measured in minutes. Water drains from the fountain at a constant rate of 1.5liters per minute. At time t equals 0, the fountain contains 48 liters of water.

How much water flows into the fountain during the time interval 0 less or equal than t less or equal than 12?

How did you do?

4b
Sme Calculator
2 marks

During the time interval 0 less or equal than t less or equal than 12, how many liters of water are in the fountain at t equals 12?

How did you do?

4c
Sme Calculator
1 mark

For t greater than 12, at what time t does the fountain run out of water?

How did you do?

4d
Sme Calculator
4 marks

For 0 less or equal than t less or equal than 12, at what time t is the amount of water in the fountain at a minimum? To the nearest liter, find the minimum volume of water in the fountain at this time. Justify your answer.

How did you do?

5a
Sme Calculator
3 marks

For t greater or equal than 0 a particle P moves along a straight line. The velocity of P at time t is given by v subscript P left parenthesis t right parenthesis equals 5 plus 3 space sin space open parentheses t over 2 close parentheses. The particle P is at position x equals negative 4 at time t equals 2, where x is the distance in meters and t is the time in seconds.

Find the position of the particle at t equals 0.

How did you do?

5b
Sme Calculator
4 marks

A second particle Q has position x equals 11 at t equals 0. Q travels on the same straight line as P at a constant velocity that is equal to the average velocity of particle P in the time 0 less or equal than t less or equal than 12. What is the distance between particles Q and P at t equals 2

How did you do?

6a
Sme Calculator
2 marks

People enter a line for an escalator at a rate modeled by the function r given by

r \left(t\right) = \left\{\begin{matrix} 44 \left(\frac{t}{100}\right)^{3} \left(1 - \frac{t}{300}\right)^{7} & \text{for } 0 \leq t \leq 300 \\ 0 & \text{for } t > 300 \end{matrix}\right.

where r \left(t\right) is measured in people per second and t is measured in seconds. As people get on the escalator, they exit the line at a constant rate of 0.7 person per second. There are 20 people in line at time t = 0.

(Note: Your calculator should be in radian mode.)

How many people enter the line for the escalator during the time interval 0 \leq t \leq 300?

How did you do?

6b
Sme Calculator
2 marks

During the time interval 0 \leq t \leq 300, there are always people in line for the escalator. How many people are in line at time t = 300?

How did you do?

6c
Sme Calculator
1 mark

For t > 300, what is the first time t that there are no people in line for the escalator?

How did you do?

7
Sme Calculator
3 marks

A second particle, Q, also moves along the x-axis so that its velocity for 0 \leq t \leq 4 is given by v_Q(t) = 45\sqrt{t} \cdot \text{cos} \; (0.063t^2) meters per hour. Find the time interval during which the velocity of particle Q is at least 60 meters per hour. Find the distance traveled by particle Q during the interval when the velocity of particle Q is at least 60 meters per hour.

How did you do?