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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Definite Integrals in Context

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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