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

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

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Definite Integrals in Context

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1
Sme Calculator2 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.

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2
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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.

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3
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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.

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4a
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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.

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4b
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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.?

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5
Sme Calculator3 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.

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1a
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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.

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1b
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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.

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2a
Sme Calculator3 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.

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2b
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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.

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3
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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?

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4a
Sme Calculator2 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?

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

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5
Sme Calculator4 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.

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1a
Sme Calculator1 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.

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1b
Sme Calculator2 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.

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1c
Sme Calculator4 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.

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2
Sme Calculator4 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.

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3a
Sme Calculator2 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?

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3b
Sme Calculator3 marks

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

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3c
Sme Calculator3 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.

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4a
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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?

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4b
Sme Calculator2 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?

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4c
Sme Calculator1 mark

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

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4d
Sme Calculator4 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.

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5a
Sme Calculator3 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.

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5b
Sme Calculator4 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

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