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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
Sme Calculator3 marks

t (minutes)

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15

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90

v open parentheses t close parentheses (inches per minute)

1.4

1.8

0.7

1.1

1.9

2.1

1.5

A snail is traveling in one direction along a straight line with velocity v open parentheses t close parentheses, in inches per minute at time t minutes, where v is a continuous function of t. Selected values of v open parentheses t close parentheses for 0 less or equal than t less or equal than 90 are shown in the table above.

Use a midpoint Riemann sum with three subintervals of equal length and values from the table to approximate integral subscript 0 superscript 90 v open parentheses t close parentheses d t. Show the computations that lead to your answer. Using correct units, explain the meaning of integral subscript 0 superscript 90 v open parentheses t close parentheses d t in terms of the snail's travel.

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2a
Sme Calculator1 mark
Graph of function f shows a curve peaking at y=7 around x=1, with x and y axes ranging between -4 and 4, and -1 and 7 respectively.

The graph of a differentiable function f on the closed interval [-4, 4] is shown in the figure above. The graph of f has a local maximum at the point (1, 7). Let g open parentheses x close parentheses equals 3 plus integral subscript 1 superscript x f open parentheses t close parentheses d t for negative 4 less or equal than x less or equal than 4.

Find a trapezoidal approximation of integral subscript negative 4 end subscript superscript 4 f open parentheses x close parentheses d x using four subintervals of length increment x equals 2.

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2b
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Find g open parentheses 1 close parentheses, g apostrophe open parentheses 1 close parentheses and g apostrophe apostrophe open parentheses 1 close parentheses.

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3
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t

(seconds)

0

30

60

80

100

120

f open parentheses t close parentheses

(liters per second)

0

0.008

0.012

0.008

0.004

0

A customer is filling up a bottle of olive oil from a large container in a health food shop. The rate of flow of the olive oil is modeled by a differentiable function f, where f open parentheses t close parentheses is measured in liters per second and t is measured in seconds since filling the bottle began. Selected values of f open parentheses t close parentheses are given in the table.

Using correct units, interpret the meaning of integral subscript 30 superscript 100 f open parentheses t close parentheses space italic d t in the context of the problem. Use a right Riemann sum with the three subintervals [30, 60], [60, 80], and [80, 100] to approximate the value of integral subscript 30 superscript 100 f open parentheses t close parentheses italic d t

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4a
Sme Calculator2 marks

x

-3

0

4

10

15

f open parentheses x close parentheses

8

6

5

1

0

Let f be a function that is twice differentiable and decreasing for all real numbers. The above table gives values of f for selected points in the interval negative 3 less or equal than x less or equal than 15.

Evaluate integral subscript negative 3 end subscript superscript 15 open parentheses 10 minus 2 f apostrophe open parentheses x close parentheses close parentheses d x. Show the work that leads to your answer.

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4b
Sme Calculator2 marks

Use a left Riemann sum with subintervals indicated by the data in the table to approximate integral subscript negative 3 end subscript superscript 15 f open parentheses x close parentheses d x. Show the work that leads to your answer.

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4c
Sme Calculator1 mark

Is your approximation in part (b) greater than or less than integral subscript negative 3 end subscript superscript 15 f open parentheses x close parentheses d x? Give a reason for your answer.

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5
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Let f be the function defined by f open parentheses x close parentheses equals sin x plus e to the power of x. Let g be the function defined by g open parentheses x close parentheses equals integral subscript 0 superscript x f open parentheses t close parentheses d t.

Find g open parentheses pi close parentheses, g apostrophe open parentheses pi close parentheses and g apostrophe apostrophe open parentheses pi close parentheses.

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1
Sme Calculator2 marks
The graph of a function f consisting of four line segments connecting the points (-4, 0) and (-1, 4), (-1, 4) and (1, 3), (1, 3) and (2, -3), and (2, -3) and (4, 0)

Let f be a continuous function defined on the closed interval negative 4 less or equal than x less or equal than 4. The graph of f, consisting of four line segments, is shown above. Let g be the function defined by g open parentheses x close parentheses equals integral subscript 0 superscript x f open parentheses t close parentheses space d t.

On what open intervals is the graph of g concave up? Give a reason for your answer.

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2a
Sme Calculator1 mark

bold italic x

0

1

3

6

Error converting from MathML to accessible text.

12

10

7

11

Error converting from MathML to accessible text.

-1

-2

3

1

f is a differentiable function. The table shown gives values of the function f and its first derivative at selected values of x.

Let g be the function defined by g open parentheses x close parentheses equals 2 x cubed plus integral subscript 0 superscript x f to the power of apostrophe open parentheses t close parentheses space d t. Find g open parentheses 3 close parentheses. Show the work that leads to your answer.

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2b
Sme Calculator3 marks

Is the function g defined in part (a) increasing, decreasing, or neither at x equals 3? Justify your answer.

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3a
Sme Calculator2 marks

bold italic r

(meters)

0

0.5

1

2

3

bold italic f stretchy left parenthesis r stretchy right parenthesis

(kilograms per square meter)

2

5

9

12

14

The density of algae on the surface of a circular garden pond at a distance of r meters from the center of the pond is given by an increasing, differentiable function f, where f open parentheses r close parentheses is measured in kilograms per square meter. Values of f open parentheses r close parentheses for selected values of r are given in the table above.

The total mass, in kilograms, of algae in the pond is given by the integral expression space 2 pi integral subscript 0 superscript 3 r f open parentheses r close parentheses space d r. Approximate the value of space 2 pi integral subscript 0 superscript 3 r f open parentheses r close parentheses space d r space using a right Riemann sum with the four subintervals indicated by the data in the table.

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3b
Sme Calculator2 marks

Is the approximation found in part (a) an overestimate or underestimate of the total mass of algae in the pond? Explain your reasoning.

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4a
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bold italic t (minutes)

0

3

7

12

15

bold italic M stretchy left parenthesis t stretchy right parenthesis (degrees Fahrenheit)

212.0

165.8

125.9

97.3

87.0

The temperature of coffee in a mug at time t is modeled by a strictly decreasing, twice-differentiable function M, where M open parentheses t close parentheses is measured in degrees Fahrenheit and t is measured in minutes. At time t equals 0, the temperature of the coffee is 212 degree straight F. The mug of coffee is then left to cool, beginning at time t equals 0. Values of M open parentheses t close parentheses at selected times t for the first 15 minutes are given in the table above.

Use the data in the table to evaluate integral subscript 0 superscript 15 M to the power of apostrophe open parentheses t close parentheses space d t. Using correct units, interpret the meaning of integral subscript 0 superscript 15 M to the power of apostrophe open parentheses t close parentheses space d t in the context of this problem.

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4b
Sme Calculator3 marks

For 0 less or equal than t less or equal than 15, the average temperature of the water in the mug is 1 over 15 integral subscript 0 superscript 15 M open parentheses t close parentheses space d t. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate 1 over 15 integral subscript 0 superscript 15 M open parentheses t close parentheses space d t. Does this approximation overestimate or underestimate the average temperature of the coffee over these 15 minutes? Explain your reasoning.

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4c
Sme Calculator2 marks

For 15 less or equal than t less or equal than 20, the function M that models the coffee temperature has first derivative given by M to the power of apostrophe open parentheses t close parentheses equals negative 18.45 e to the power of negative 0.125 t end exponent. Based on the model, what is the temperature of the coffee at time t equals 20?

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5
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Let f be the function defined by f open parentheses x close parentheses equals integral subscript 1 superscript x open parentheses 1 over t plus t close parentheses squared d t for x greater or equal than 1.

Find f open parentheses 3 close parentheses, f apostrophe open parentheses 3 close parentheses and f apostrophe apostrophe open parentheses 3 close parentheses.

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1
Sme Calculator3 marks

bold italic x

0

10

20

30

40

Error converting from MathML to accessible text.

2

6

7

9

11

f is a continuous function which is increasing on the interval open square brackets 0 comma space 40 close square brackets. The table shown gives values of the function f at selected values of x.

Using relevant approximations with all the data in table, find an interval of shortest width which contains the exact value of integral subscript 0 superscript 40 f open parentheses x close parentheses d x. Justify why your interval is valid.

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2
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A graph of f', the derivative of the function f, consisting of a line segment connecting points (0, 5) and (2, 5), another line segment connecting points (2, 5) and (5, 0), and a semicircle of radius 2 below the x-axis connecting points (5, 0) and (9, 0)

The function f is defined on the closed interval open square brackets 0 comma space 9 close square brackets and satisfies f open parentheses 3 close parentheses equals 2. 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 above.

Find the absolute minimum value of f on the closed interval open square brackets 0 comma space 9 close square brackets. Justify your answer.

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3a
Sme Calculator3 marks
Graph of function f with shaded region R. Points marked at (-3, 1) and (3, -2). The region R is in the second quadrant and is bounded by the graph of f, the vertical line x=-3 and the coordinate axes. From 0 to 4 the graph is linear.

The graph of the differentiable function f, shown for negative 3 less or equal than x less or equal than 4, is linear for 0 less or equal than x less or equal than 4. Let R be the region in the second quadrant bounded by the graph of f, the vertical line x equals negative 3 and the x- and y- axes. The area of region R is 12.

The function g is defined by g open parentheses x close parentheses equals integral subscript 0 superscript x f open parentheses t close parentheses d t. Find the values of g open parentheses negative 3 close parentheses, g open parentheses 2 close parentheses and g open parentheses 3 close parentheses.

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3b
Sme Calculator4 marks

The function h is defined by h open parentheses x close parentheses equals integral subscript negative 3 end subscript superscript x f apostrophe open parentheses t close parentheses d t. Find the values of h open parentheses 3 close parentheses, h apostrophe open parentheses 3 close parentheses and h apostrophe apostrophe open parentheses 3 close parentheses.

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4a
Sme Calculator3 marks

t (seconds)

0

5

10

20

30

50

h apostrophe open parentheses t close parentheses (feet per second)

2

3

5

11

19

41

A person is on a roller coaster ride which rises vertically rapidly. The height of the person from the ground is modeled by the twice-differentiable function h open parentheses t close parentheses for 0 less or equal than t less or equal than 50, where t is measured in seconds and h open parentheses t close parentheses is measured in feet. Values of h apostrophe open parentheses t close parentheses at selected values of t are shown in the table above. Initially, the person is 1 foot above the ground.

Use a trapezoidal sum with five subintervals indicated by the table to estimate the height of the person at time t equals 50 seconds.

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4b
Sme Calculator1 mark

It is known that the function h apostrophe apostrophe open parentheses t close parentheses is increasing on the interval 0 less or equal than t less or equal than 50.

Is your approximation in part (a) greater than or less than the height of the person from the ground, as predicted by the model, at time t equals 50 seconds? Give a reason for your answer.

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5a
Sme Calculator2 marks

Let f be the function defined by f open parentheses x close parentheses equals fraction numerator 4 x squared plus x plus 3 over denominator 2 x squared end fraction for x greater than 0.

Evaluate integral subscript 1 superscript 3 f open parentheses x close parentheses d x.

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5b
Sme Calculator3 marks

Let g be the function defined by g open parentheses x close parentheses equals integral subscript 1 superscript x f apostrophe apostrophe open parentheses t close parentheses d t for x greater than 0.

Evaluate integral subscript 1 superscript 3 g open parentheses x close parentheses d x.

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