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

Riemann Sums & Definite Integrals (College Board AP® Calculus BC): Exam Questions

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Riemann Sums & Definite Integrals

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1
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integral subscript 1 superscript e open parentheses fraction numerator 3 x squared minus 1 over denominator x end fraction close parentheses space d x equals

  • 3 e minus 1 over e

  • 3 e squared minus e

  • fraction numerator 3 e squared over denominator 2 end fraction minus 5 over 2

  • 3 e squared minus 1

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2
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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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3
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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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4
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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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5
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Line graph with four points (-2, 3), (-1, 1), (1, 2) and (3, -2) connected by line segments. They intersect the axes at (0, 3/2) and (2, 0). Title below reads "Graph of f".

The graph of the piecewise linear function f is shown in the figure above. If g open parentheses x close parentheses equals integral subscript negative 1 end subscript superscript x f open parentheses t close parentheses space d t, which of the following values is the greatest?

  • g open parentheses negative 2 close parentheses

  • g open parentheses 0 close parentheses

  • g open parentheses 2 close parentheses

  • g open parentheses 3 close parentheses

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6
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integral subscript 1 superscript 2 1 over x cubed d x equals

  • negative 15 over 4

  • negative 3 over 8

  • 3 over 8

  • 3 ln 2

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7
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A graph showing a sinusoidal wave through the origin, extending horizontally from 0 to 8, labelled as "Graph of f" with x and y axes. The graph starts as positive and crossed the x-axis at 1, 3, 5 and 7.

The function f is continuous on the interval open square brackets 0 comma space 8 close square brackets. The figure above shows the graph of f. Let g be the function defined by g open parentheses x close parentheses equals integral subscript 4 superscript x f open parentheses t close parentheses d t. On what intervals is g decreasing?

  • open square brackets 0 comma space 2 close square brackets and open square brackets 4 comma space 6 close square brackets only

  • open square brackets 0 comma space 1 close square brackets, open square brackets 3 comma space 5 close square brackets and open square brackets 7 comma space 8 close square bracketsonly

  • open square brackets 1 comma space 3 close square brackets and open square brackets 5 comma space 7 close square brackets only

  • open square brackets 2 comma space 4 close square brackets and open square brackets 6 comma space 8 close square brackets only

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8
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If integral subscript 0 superscript 10 f open parentheses x close parentheses d x equals 23 and integral subscript 15 superscript 10 f open parentheses x close parentheses d x equals negative 5, what is the value of integral subscript 0 superscript 15 f open parentheses x close parentheses d x?

  • -28

  • -18

  • 18

  • 28

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9
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x

0

1

2

3

f open parentheses x close parentheses

1.25

0.45

-2.15

-5.75

f apostrophe open parentheses x close parentheses

-1.05

-3.15

1.95

2.65

f is a function such that f apostrophe is continuous on the interval open square brackets 0 comma space 3 close square brackets. The table above gives the value of the function f and its derivative at selected values of x.

integral subscript 0 superscript 3 f apostrophe open parentheses x close parentheses d x equals

  • -7

  • -3.7

  • 3.7

  • 7

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10
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integral subscript 0 superscript pi over 3 end superscript cos x d x equals

  • negative fraction numerator square root of 3 over denominator 2 end fraction

  • negative 1 half

  • 1 half

  • fraction numerator square root of 3 over denominator 2 end fraction

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1
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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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2
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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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3
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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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4
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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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5
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If a trapezoidal sum underapproximates integral subscript 0 superscript 4 f open parentheses x close parentheses space d x, and a left Riemann sum overapproximates integral subscript 0 superscript 4 f open parentheses x close parentheses space d x, which of the following could be the graph of y equals f open parentheses x close parentheses?

  • Graph showing a decreasing, concave up curve on a coordinate plane. The x-axis ranges from 0 to 4 and the y-axis from 0 to 4.
  • Graph showing n increasing, concave down curve on a coordinate plane. The x-axis ranges from 0 to 4 and the y-axis from 0 to 4.
  • Graph showing a decreasing, concave down curve on a coordinate plane. The x-axis ranges from 0 to 4 and the y-axis from 0 to 4.
  • Graph showing a increasing, concave up curve on a coordinate plane. The x-axis ranges from 0 to 4 and the y-axis from 0 to 4.

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6
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If f open parentheses x close parentheses equals m x plus b and c less than x less than d then integral subscript c superscript d f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses d x equals

  • 0

  • d minus c

  • m open parentheses d minus c close parentheses

  • m d

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7
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Let F open parentheses x close parentheses be an antiderivative of sin x open parentheses 1 plus cos x close parentheses squared. If F open parentheses 1 close parentheses equals 2, then F open parentheses 3 close parentheses equals

  • -0.782

  • 1.218

  • 2.000

  • 3.218

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8
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For negative pi over 2 less than x less than pi over 2, if f open parentheses x close parentheses equals integral subscript x superscript 0 sec squared t d t then f apostrophe open parentheses x close parentheses equals

  • sec squared x

  • negative sec squared x

  • 1 minus sec squared x

  • negative tan x

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9
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Graph of a piecewise function with three segments, crossing points at (-2, 3), (-1, 0), (0, 1), (1, 2), (2, 0) and (3, -2)on the xy-plane.

The graph of a piece-wise linear function f is shown above for negative 2 less or equal than x less or equal than 3. What is the value of integral subscript negative 2 end subscript superscript 3 f open parentheses x close parentheses d x?

  • 1.5

  • 3.5

  • 4.5

  • 5.5

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1
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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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2
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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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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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4
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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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5
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Let f be a continuous function. If G open parentheses x close parentheses is an antiderivative for f open parentheses x close parentheses and G open parentheses 7 close parentheses equals negative 3, then G open parentheses 1 close parentheses equals

  • f to the power of apostrophe open parentheses 1 close parentheses

  • table row blank blank cell integral subscript 1 superscript 7 end cell end table table row blank blank cell f open parentheses t close parentheses space end cell end table table row blank blank d end table table row blank blank t end table

  • negative table row blank blank cell integral subscript 1 superscript 7 end cell end table table row blank blank cell open parentheses 1 half plus f open parentheses t close parentheses close parentheses space d t end cell end table

  • 3 plus table row blank blank cell integral subscript 1 superscript 7 end cell end table table row blank blank cell f open parentheses t close parentheses space end cell end table table row blank blank d end table t

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6
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kis a constant. If integral subscript 0 superscript k open parentheses 2 k x cubed plus 5 x to the power of 4 close parentheses d x equals 48, then k equals

  • negative 2

  • negative fifth root of 24

  • fifth root of 24

  • 2

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7
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integral subscript 0 superscript 5 open vertical bar x minus 4 close vertical bar d x equals

  • negative 17 over 2

  • negative 15 over 2

  • 15 over 2

  • 17 over 2

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8
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Graph of function f, showing an upward curve starting at point (0,1), with x-axis labelled 0 to 4. The function is increasing and concave up.

The graph of the function f is shown above for 0 less or equal than x less or equal than 4. Which of the following has the greatest value?

  • Left Riemann sum approximation of integral subscript 0 superscript 4 f open parentheses x close parentheses d x with 4 subintervals of equal length

  • Midpoint Riemann sum approximation of integral subscript 0 superscript 4 f open parentheses x close parentheses d x with 4 subintervals of equal length

  • Right Riemann sum approximation of integral subscript 0 superscript 4 f open parentheses x close parentheses d x with 4 subintervals of equal length

  • Trapezoidal sum approximation of integral subscript 0 superscript 4 f open parentheses x close parentheses d x with 4 subintervals of equal length

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9
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fraction numerator d over denominator d x end fraction open parentheses integral subscript 0 superscript x cubed end superscript fraction numerator 1 over denominator t to the power of 4 plus 1 end fraction d t close parentheses equals

  • fraction numerator 3 x squared over denominator x to the power of 4 plus 1 end fraction

  • fraction numerator 1 over denominator x to the power of 12 plus 1 end fraction

  • fraction numerator 3 x squared over denominator x to the power of 12 plus 1 end fraction

  • fraction numerator 6 x to the power of 5 over denominator x to the power of 12 plus 1 end fraction

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10
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Let f be the function given by f left parenthesis x right parenthesis equals integral subscript 0 superscript x cos open parentheses t squared close parentheses d t for negative 1 less or equal than x less or equal than 1.5. On which of the following intervals is f increasing?

  • negative 1 less or equal than x less or equal than 0

  • negative 1 less or equal than x less or equal than 1.253

  • 0 less or equal than x less or equal than 1.5

  • 1.355 less or equal than x less or equal than 1.5

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