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

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

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Definite Integrals in Context

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1
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The acceleration, in feet per second per second, of a particle moving in a straight line is given by the function a open parentheses t close parentheses equals e to the power of x cube root of x minus 2 x. What is the average acceleration of the particle from time t equals 0 to time t equals 4?

  • 15.241 ft/sec2

  • 23.241 ft/sec2

  • 30.481 ft/sec2

  • 60.962 ft/sec2

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2
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A particle moves along the x-axis with velocity given by v open parentheses t close parentheses equals 6 t squared minus 2 t for time t greater or equal than 0. If the particle starts at open parentheses 0 comma space 0 close parentheses, what is the position of the particle at time t equals 3?

  • 34

  • 45

  • 48

  • 63

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3
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What is the average value of y equals fraction numerator sin space x over denominator x squared minus 5 end fraction on the closed interval open square brackets negative 2 comma space 1 close square brackets?

  • negative 0.137

  • 0.137

  • 0.206

  • 0.412

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4
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A diagram of a park next to a highway. The park is rectangular with a horizontal length of 7 km and a vertical length of 3 km. The highway runs alongside the left hand vertical edge of the park.

A rectangular park is located beside a highway as shown in the figure above. A population of ducks lives in the park and at any point along a strip x meters from the the edge of the highway are f left parenthesis x right parenthesis ducks per square meter. Which of the following expressions gives the total population of ducks in the park?

  • integral subscript 0 superscript 2 f open parentheses x close parentheses space d x

  • 6 integral subscript 0 superscript 2 f open parentheses x close parentheses space d x

  • integral subscript 0 superscript 6 f open parentheses x close parentheses space d x

  • 2 integral subscript 0 superscript 6 f open parentheses x close parentheses space d x

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5
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The water level in a tank is given by the function integral subscript 0 superscript t r open parentheses t close parentheses italic d t where r open parentheses t close parentheses equals 2 t cubed minus 9 t squared plus 3 for 0 less or equal than t less or equal than 6. During which of the following intervals is the water level increasing?

  • 0 less or equal than t less or equal than 0.621 and 4.423 less or equal than t less or equal than 6

  • 0.621 less or equal than t less or equal than 4.423

  • 3 less or equal than t less or equal than 6

  • 5.423 less or equal than t less or equal than 6

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6
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The graph below gives the velocity, v, in ft/sec, of an object for 0 less or equal than t less or equal than 6, where t is the time in seconds. Of the following, which is the best estimate of the distance traveled by the object for 0 less or equal than t less or equal than 6?

Graph showing a curve with velocity v on the vertical axis and time t on the horizontal axis. Curve starts at v = 40, t = 0 and decreases to t=6, v=-140. Also passes through approximately t=2.8, v=0.

  • -170 ft

  • 280 ft

  • 480 ft

  • 540 ft

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1
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A machine is filling a bottle of vinegar at a rate of r open parentheses t close parentheses equals 12.8 e to the power of negative 0.4 t end exponent ounces per second, where t is measured in seconds. How many ounces of vinegar are put into the bottle from time t equals 0 to time t equals 8?

  • 0.522

  • 2.584

  • 12.800

  • 30.696

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2
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A particle moves along the x-axis. The velocity of the particle at time t is 12 t minus 3 t squared. What is the total distance traveled by the particle from time t equals 0 to t equals 2?

  • 24

  • 16

  • 15

  • 12

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3
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A particle moves along the x-axis. The velocity of the particle at time t is given by v open parentheses t close parentheses, and the acceleration of the particle at time t is given by a open parentheses t close parentheses. Which of the following gives the average velocity of the particle from time t equals 0 to time t equals 12?

  • fraction numerator v open parentheses 12 close parentheses minus v open parentheses 0 close parentheses over denominator 12 end fraction

  • 1 half integral subscript 0 superscript 12 v open parentheses t close parentheses space d t

  • 1 over 12 integral subscript 0 superscript 12 v open parentheses t close parentheses space d t

  • 1 over 12 integral subscript 0 superscript 12 open vertical bar v open parentheses t close parentheses close vertical bar space d t

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4
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A particle moves along a line so that its acceleration for t greater or equal than 0 is given by a open parentheses t close parentheses equals fraction numerator t squared plus 1 over denominator square root of t cubed plus 7 end root end fraction. If the particle's velocity at t equals 0 is 12, what is the velocity of the particle at t equals 5?

  • 7.032

  • 13.119

  • 19.032

  • 25.119

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5
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Line graph showing a cubic feet per minute flow rate over 10 minutes. The flow starts at 20, peaks near 55 at 5 minutes, then returns to 20 by 10 minutes.

The flow of natural gas, in cubic feet per minute, through a pipe is given by the graph shown above. Of the following, which best approximates the total number of cubic feet of natural gas that passed through the pipe during the 10 minutes shown on the graph?

  • 140

  • 290

  • 370

  • 500

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6
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A particle with velocity at any time t given by v open parentheses t close parentheses equals e to the power of 2 t end exponent moves in a straight line. How far does the particle move from t equals 0 to t equals 4?

  • 1 half open parentheses e to the power of 8 minus 1 close parentheses

  • e to the power of 8 minus 1

  • 2 open parentheses e to the power of 8 minus 1 close parentheses

  • 1 half e to the power of 8

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7
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What is the average value of y for the part of the curve 4 x minus x squared which is in the first quadrant?

  • negative 2

  • 8 over 3

  • 16 over 3

  • 32 over 3

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8
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A cup of coffee, initially at a temperature of 200 degrees Fahrenheit (°F), is placed in a 60 °F room at time t equals 0 minutes. The temperature of the cup of coffee is changing at a rate of negative 50 e to the power of negative 0.2 t end exponent degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the cup of coffee at time t equals 3 minutes?

  • 87 °F

  • 113 °F

  • 147 °F

  • 173 °F

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1
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On the closed interval open square brackets 1 comma space 5 close square brackets, which of the following could be the graph of a function g with the property that 1 fourth integral subscript 1 superscript 5 g open parentheses x close parentheses space d t equals 3 over 2 ?

  • A graph of a function that is a horizontal line going from (1, 2.5) to (5, 2.5).
  • A graph of a function that is a straight line going from (2, 3) to (5, 0).
  • A graph of a function that is made up of a straight line segment going from (1, 0) to (3, 3) and a second segment going from (3, 3) to (5, 0).
  • A graph of a function which is a smooth curve that starts at (1, 3) and goes down (3, 0) and back up to (5, 3).

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2
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A particle moves along the x-axis so that at any time t greater than 0, its acceleration is given by a open parentheses t close parentheses equals ln space open parentheses 3 to the power of t plus 2 close parentheses. If the velocity of the particle is negative 1.5 at time t equals 2, then the velocity of the particle at time t equals 4 is

  • 3.313

  • 4.419

  • 5.261

  • 6.761

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3
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An object traveling in a straight line has position x open parentheses t close parentheses at time t, where time is measured in seconds. If the velocity of the object is v left parenthesis t right parenthesis equals cube root of 2 t squared plus t end root minus 5 what is the distance traveled by the object in the first 18 seconds?

  • 3.733

  • 5.451

  • 20.183

  • 34.915

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4
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What is the average value of y equals x times cube root of x squared minus 1 end root on the interval open square brackets 0 comma space 3 close square brackets?

  • 15 over 8

  • 45 over 8

  • 15 over 4

  • 45 over 4

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5
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Water is pumped into a tank at a rate of r open parentheses t close parentheses equals x open parentheses fraction numerator 5 over denominator 3 fourth root of x cubed end root end fraction close parentheses gallons per minute, where t is the number of minutes since the pump was turned on. If the tank contained 70 gallons of water when the pump was turned on, how much water, to the nearest gallon, is in the tank after 81 minutes?

  • 56 gallons

  • 75 gallons

  • 324 gallons

  • 394 gallons

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