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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 6: Integration & Accumulation of ChangeRiemann Sums & Definite Integrals

Riemann Sums & Definite Integrals (College Board AP® Calculus AB)

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Riemann Sums & Definite Integrals

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Riemann Sums & Definite Integrals

1
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integral subscript 0 superscript x cos t space d t equals

  • cos x

  • cos x minus 1

  • sin x

  • sin x minus 1

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2
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If f is a function such that its derivative f to the power of apostrophe is continuous on open square brackets 0 comma space p close square brackets, then integral subscript 0 superscript p f to the power of apostrophe open parentheses x close parentheses space d x equals

  • f open parentheses p close parentheses minus f open parentheses 0 close parentheses

  • open vertical bar f open parentheses p close parentheses minus f open parentheses 0 close parentheses close vertical bar

  • f open parentheses p close parentheses

  • f open parentheses x close parentheses plus C

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3
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t (hours)

3

7

10

14

R(t)

(liters/hour)

5.2

5.0

4.6

4.2

A tank contains 450 liters of water at time t equals 3 hours. Water is being pumped out of the tank at a rate R open parentheses t close parentheses, where R open parentheses t close parentheses is measured in liters per hour, and t is measured in hours. Selected values of R open parentheses t close parentheses are given in the table above. Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the number of liters of water that are in the tank at time t equals 14 hours?

  • 50.6

  • 54.2

  • 397.6

  • 399.4

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4
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x

0

2

6

9

f(x)

20

10

50

70

The function f is continuous on the closed interval open square brackets 0. space 9 close square brackets and has values that are given in the table above. Using a trapezoidal sum with three subintervals and the values from the table, what is the approximation of integral subscript 0 superscript 9 f open parentheses x close parentheses space d x?

  • 250

  • 290

  • 330

  • 370

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5
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If F open parentheses x close parentheses equals integral subscript 0 superscript x square root of t cubed plus 9 end root space d t, then F to the power of apostrophe open parentheses 3 close parentheses equals

  • -6

  • -3

  • 3

  • 6

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6
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What are all the values of k for which integral subscript 4 superscript k x squared space d x equals 0?

  • -4

  • 0

  • 4

  • -4 and 4

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7
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If integral subscript 3 superscript 12 f open parentheses x close parentheses space d x equals 7 and integral subscript 6 superscript 3 f open parentheses x close parentheses space d x equals 2, then integral subscript 6 superscript 12 f open parentheses x close parentheses space d x equals

  • -5

  • 5

  • 9

  • 10

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8
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Which of the following is equivalent to the definite integral integral subscript negative 2 end subscript superscript 3 x squared space d x?

  • limit as n rightwards arrow infinity of sum from k equals 1 to n of 3 over n open parentheses fraction numerator 3 k over denominator n end fraction close parentheses squared

  • limit as n rightwards arrow infinity of sum from k equals 1 to n of 5 over n open parentheses fraction numerator 5 k over denominator n end fraction close parentheses squared

  • limit as n rightwards arrow infinity of sum from k equals 1 to n of 3 over n open parentheses fraction numerator 3 k over denominator n end fraction minus 2 close parentheses squared

  • limit as n rightwards arrow infinity of sum from k equals 1 to n of 5 over n open parentheses fraction numerator 5 k over denominator n end fraction minus 2 close parentheses squared

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9
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Graph of a continuous function f which is positive between x=0 and x=5, crosses the x-axis at x=5, and is negative between x=5 and x=9

The graph of a differentiable function f is shown above. If h open parentheses x close parentheses equals integral subscript 0 superscript x f open parentheses t close parentheses space d t, which of the following is true?

  • h open parentheses 5 close parentheses less than h to the power of apostrophe open parentheses 5 close parentheses less than h to the power of apostrophe apostrophe end exponent open parentheses 5 close parentheses

  • h to the power of apostrophe open parentheses 5 close parentheses less than h open parentheses 5 close parentheses less than h to the power of apostrophe apostrophe end exponent open parentheses 5 close parentheses

  • h to the power of apostrophe apostrophe end exponent open parentheses 5 close parentheses less than h open parentheses 5 close parentheses less than h to the power of apostrophe open parentheses 5 close parentheses

  • h to the power of apostrophe apostrophe end exponent open parentheses 5 close parentheses less than h to the power of apostrophe open parentheses 5 close parentheses less than h open parentheses 5 close parentheses

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10
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A graph of a straight line, starting at a positive value of y when x=0, then sloping downwards to cross the x-axis and continue on into negative values of y

The figure above shows the graph of f. If f open parentheses x close parentheses equals integral subscript 3 superscript x g open parentheses t close parentheses space d t, which of the following could be the graph of y equals g open parentheses x close parentheses?

  • A graph of a constant function (horizontal line) above the x-axis
  • A graph of a constant function (horizontal line) below the x-axis
  • A graph of a straight line, starting at the origin and increasing to the right with a positive slope
  • A graph of a quadratic function, starting at the origin and going down below the x-axis, before curving back upwards, crossing the x-axis, and continuing to increase

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11
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If f to the power of apostrophe open parentheses x close parentheses less than 0 for all real numbers x and integral subscript 3 superscript 6 f open parentheses t close parentheses space d t equals 0, which of the following could be a table of values for the function f?

  • x

    f(x)

    3

    5

    5

    2

    6

    0

  • x

    f(x)

    3

    5

    5

    -1

    6

    3

  • x

    f(x)

    3

    5

    5

    8

    6

    9

  • x

    f(x)

    3

    5

    5

    1

    6

    -2

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