Sample Test

Definite Integrals and Accumulation

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Material Overview

Net change, signed area, average value, and units

Subject Mathematics
Subject Area Calculus

Grade/Level 12th Grade

Type Test

Difficulty Challenging

Learning Setting Classroom

Back to Samples Create Your Own

Teacher Context

Teacher Lesson Scope

What Students Should Know

Students must understand the fundamental relationship between differentiation and integration, specifically how integrating a rate of change yields a net change in the accumulated quantity. They should be comfortable working with rate functions presented algebraically, graphically, and numerically. Students also need to understand how the sign of a function relates to the area bounded by its graph and the horizontal axis.

Key Skills

  • Distinguish between net change (displacement) and total accumulated magnitude (total distance).
  • Calculate the average value of a function over a closed interval using definite integrals.
  • Set up and solve accumulation equations that incorporate an initial condition.
  • Use units of measurement to interpret the physical meaning of definite integrals in real-world contexts.
  • Evaluate definite integrals from equations, geometric regions, and Riemann sum tables.

Important Vocabulary

  • Net Change: The net difference between the starting value and the ending value of a quantity over a specific interval, calculated by integrating its rate of change.
  • Signed Area: The geometric area of a region between a function and the horizontal axis, where regions above the axis contribute positively and regions below the axis contribute negatively.
  • Average Value: The horizontal line height that represents the average value of a continuous function over an interval, given by dividing the definite integral over that interval by the interval's width.
  • Total Distance: The total path length traveled by an object, regardless of direction, found by integrating the absolute value of the velocity function.
  • Accumulation Function: A function that represents the total accumulated amount of a quantity, often expressed as an initial value plus a definite integral of a rate function.

Assessment Boundaries

This assessment focuses strictly on the physical interpretation and calculation of net change, signed area, and average value. It contains a mix of algebraic, graphical, and tabular questions. Transcendental functions are restricted to basic exponential and trigonometric expressions. Numerical approximation is limited to Riemann sums from data tables.

Printable Student Copy

Student Version

Name:
Date:

Welcome to the Net Change, Signed Area, Average Value, and Units Test. This exam will evaluate your conceptual and computational understanding of integration as accumulation. Please read each question carefully, show all your steps clearly in the workspace provided for the short answer questions, and work individually to complete each section.

Part I: Multiple Choice Questions (Questions 1-8)

Select the single best answer for each question.

  1. A particle moves along a straight line with a velocity function given by v(t) = 3t² - 12t for 0 ≤ t ≤ 5, where t is in seconds and velocity is in meters per second. What is the difference between the total distance traveled by the particle and its net change in position over the interval [0, 5]?
    A. 14 meters
    B. 39 meters
    C. 50 meters
    D. 64 meters

  2. What is the average value of the function f(x) = exp(-0.5x) over the interval [0, 4]?
    A. 2 - 2 × exp(-2)
    B. 0.5 - 0.5 × exp(-2)
    C. 1 - exp(-2)
    D. 0.25 - 0.25 × exp(-2)

  3. Water leaks from a municipal storage tank at a rate modeled by C(t) liters per hour, where t is measured in hours since midnight. Which of the following is the best physical interpretation of the expression (1/6) × ∫[2, 8] C(t) dt?
    A. The total volume of water in liters that leaked from the tank between 2:00 AM and 8:00 AM.
    B. The instantaneous rate in liters per hour at which water is leaking from the tank at 5:00 AM.
    C. The average rate of water leakage in liters per hour between 2:00 AM and 8:00 AM.
    D. The total change in the rate of leakage in liters per hour squared between 2:00 AM and 8:00 AM.

  4. Oil flows through a pipeline at a rate of r(t) gallons per minute, where t is in minutes. Selected values of r(t) are given in the table below:

t (min) 0 3 6 9 12
r(t) (gal/min) 10 13 15 12 8

Using a Right Riemann Sum with four subintervals of equal length, what is the estimated total number of gallons of oil that flowed through the pipeline over the 12-minute period?
A. 114 gallons
B. 144 gallons
C. 147 gallons
D. 150 gallons

  1. A water holding tank contains 50 gallons of water at time t = 0 hours. Water is pumped into the tank at a rate of E(t) = 8t gallons per hour, and water leaks out of the tank at a rate of L(t) = t² gallons per hour. How many gallons of water are in the tank at t = 6 hours?
    A. 72 gallons
    B. 122 gallons
    C. 144 gallons
    D. 194 gallons

  2. Let the function g be defined by g(x) = ∫[1, x²] cos(t) dt. What is the rate of change of g(x) with respect to x at the point where x = √(π)?
    A. -2√(π)
    B. -√(π)
    C. 0
    D. 2√(π)

  3. A bee colony population is modeled by a rate of change P'(t) = 120 - 30t bees per week, where t is the time in weeks for 0 ≤ t ≤ 5. If the initial population at t = 0 is 1000 bees, what is the maximum population reached by the colony during the 5-week period?
    A. 1000 bees
    B. 1225 bees
    C. 1240 bees
    D. 1350 bees

  4. A continuous function f(x) is defined on the interval [0, 8]. The regions bounded by the graph of f(x) and the x-axis have the following areas:

  5. From x = 0 to x = 3, f(x) is above the x-axis with an area of 12.
  6. From x = 3 to x = 6, f(x) is below the x-axis with an area of 4.
  7. From x = 6 to x = 8, f(x) is above the x-axis with an area of 8.
    What is the average value of f(x) over the entire interval [0, 8]?
    A. 2
    B. 3
    C. 4
    D. 24

Part II: Short Answer Questions (Questions 9-14)

Show all necessary formulas, integration steps, and evaluations in the workspaces provided.

  1. The graph of the rate of change of a quantity Q(t), given by f'(t) in units per minute, is defined on the interval [0, 6] and consists of three line segments:
  2. Segment A: a line segment from (0, 4) to (2, 0)
  3. Segment B: a line segment from (2, 0) to (4, -4)
  4. Segment C: a line segment from (4, -4) to (6, 0)

Part A: Calculate the net change in Q(t) over the interval [0, 6].
Net Change:

Part B: Calculate the total accumulated change in magnitude of Q(t) over the interval [0, 6] (the integral of the absolute value of the rate of change).
Total Accumulated Change:

  1. An object moves along a straight coordinate line with a velocity function given by v(t) = cos(π × t) meters per second over the time interval [0, 2] seconds.

Part A: Find the average velocity of the object over the interval.
Average Velocity:

Part B: Find the average speed of the object over the interval.
Average Speed:

  1. A hot-air balloon's altitude is changing. The rate of change of its height is modeled by the function H'(t) = 30 × √(t) - 10t meters per minute for 0 ≤ t ≤ 4 minutes. At t = 0, the balloon is at an initial height of 150 meters.

Part A: Write an expression for the height function H(t) for any time t on the interval.
H(t) =

Part B: Determine the height of the balloon at t = 4 minutes.
Height at t = 4:

  1. A chemical pollutant is being filtered and removed from an industrial lake at a rate modeled by R(t) = 50 × exp(-0.1t) kilograms per day, where t is the number of days since the filtration system began operating.

Part A: Calculate the total mass of the pollutant removed from the lake during the first 10 days of operation. Round your final answer to the nearest thousandth of a kilogram.
Total Mass Removed:

Part B: State the units of your answer and interpret what this value represents in terms of the lake's environmental remediation.

  1. The temperature of a metal bar of length 3 meters depends on the distance x from its left end. The temperature is modeled by the function T(x) = 40 + 15x² degrees Celsius for 0 ≤ x ≤ 3. Determine the average temperature of the entire metal bar.
    Average Temperature:
  1. The acceleration of water flow rate into a reservoir is modeled by f''(t) = 6t - 4 gallons per minute squared, where t is measured in minutes. At time t = 0, the initial rate of water flow into the reservoir is f'(0) = 10 gallons per minute, and the initial volume of water stored in the reservoir is f(0) = 100 gallons.

Part A: Find the equation for the water flow rate f'(t) at any time t.
f'(t) =

Part B: Find the equation for the volume of water f(t) in the reservoir at any time t.
f(t) =

Teacher Copy

Answer Key

Part I: Multiple Choice Questions (Questions 1-8)

Select the single best answer for each question.

  1. A particle moves along a straight line with a velocity function given by v(t) = 3t² - 12t for 0 ≤ t ≤ 5, where t is in seconds and velocity is in meters per second. What is the difference between the total distance traveled by the particle and its net change in position over the interval [0, 5]?
    A. 14 meters
    B. 39 meters
    C. 50 meters
    D. 64 meters

  2. What is the average value of the function f(x) = exp(-0.5x) over the interval [0, 4]?
    A. 2 - 2 × exp(-2)
    B. 0.5 - 0.5 × exp(-2)
    C. 1 - exp(-2)
    D. 0.25 - 0.25 × exp(-2)

  3. Water leaks from a municipal storage tank at a rate modeled by C(t) liters per hour, where t is measured in hours since midnight. Which of the following is the best physical interpretation of the expression (1/6) × ∫[2, 8] C(t) dt?
    A. The total volume of water in liters that leaked from the tank between 2:00 AM and 8:00 AM.
    B. The instantaneous rate in liters per hour at which water is leaking from the tank at 5:00 AM.
    C. The average rate of water leakage in liters per hour between 2:00 AM and 8:00 AM.
    D. The total change in the rate of leakage in liters per hour squared between 2:00 AM and 8:00 AM.

  4. Oil flows through a pipeline at a rate of r(t) gallons per minute, where t is in minutes. Selected values of r(t) are given in the table below:

t (min) 0 3 6 9 12
r(t) (gal/min) 10 13 15 12 8

Using a Right Riemann Sum with four subintervals of equal length, what is the estimated total number of gallons of oil that flowed through the pipeline over the 12-minute period?
A. 114 gallons
B. 144 gallons
C. 147 gallons
D. 150 gallons

  1. A water holding tank contains 50 gallons of water at time t = 0 hours. Water is pumped into the tank at a rate of E(t) = 8t gallons per hour, and water leaks out of the tank at a rate of L(t) = t² gallons per hour. How many gallons of water are in the tank at t = 6 hours?
    A. 72 gallons
    B. 122 gallons
    C. 144 gallons
    D. 194 gallons

  2. Let the function g be defined by g(x) = ∫[1, x²] cos(t) dt. What is the rate of change of g(x) with respect to x at the point where x = √(π)?
    A. -2√(π)
    B. -√(π)
    C. 0
    D. 2√(π)

  3. A bee colony population is modeled by a rate of change P'(t) = 120 - 30t bees per week, where t is the time in weeks for 0 ≤ t ≤ 5. If the initial population at t = 0 is 1000 bees, what is the maximum population reached by the colony during the 5-week period?
    A. 1000 bees
    B. 1225 bees
    C. 1240 bees
    D. 1350 bees

  4. A continuous function f(x) is defined on the interval [0, 8]. The regions bounded by the graph of f(x) and the x-axis have the following areas:

  5. From x = 0 to x = 3, f(x) is above the x-axis with an area of 12.
  6. From x = 3 to x = 6, f(x) is below the x-axis with an area of 4.
  7. From x = 6 to x = 8, f(x) is above the x-axis with an area of 8.
    What is the average value of f(x) over the entire interval [0, 8]?
    A. 2
    B. 3
    C. 4
    D. 24

Part II: Short Answer Questions (Questions 9-14)

Show all necessary formulas, integration steps, and evaluations in the workspaces provided.

  1. The graph of the rate of change of a quantity Q(t), given by f'(t) in units per minute, is defined on the interval [0, 6] and consists of three line segments:
  2. Segment A: a line segment from (0, 4) to (2, 0)
  3. Segment B: a line segment from (2, 0) to (4, -4)
  4. Segment C: a line segment from (4, -4) to (6, 0)

Part A: Calculate the net change in Q(t) over the interval [0, 6].
Net Change:

Part B: Calculate the total accumulated change in magnitude of Q(t) over the interval [0, 6] (the integral of the absolute value of the rate of change).
Total Accumulated Change:

  1. An object moves along a straight coordinate line with a velocity function given by v(t) = cos(π × t) meters per second over the time interval [0, 2] seconds.

Part A: Find the average velocity of the object over the interval.
Average Velocity:

Part B: Find the average speed of the object over the interval.
Average Speed:

  1. A hot-air balloon's altitude is changing. The rate of change of its height is modeled by the function H'(t) = 30 × √(t) - 10t meters per minute for 0 ≤ t ≤ 4 minutes. At t = 0, the balloon is at an initial height of 150 meters.

Part A: Write an expression for the height function H(t) for any time t on the interval.
H(t) =

Part B: Determine the height of the balloon at t = 4 minutes.
Height at t = 4:

  1. A chemical pollutant is being filtered and removed from an industrial lake at a rate modeled by R(t) = 50 × exp(-0.1t) kilograms per day, where t is the number of days since the filtration system began operating.

Part A: Calculate the total mass of the pollutant removed from the lake during the first 10 days of operation. Round your final answer to the nearest thousandth of a kilogram.
Total Mass Removed:

Part B: State the units of your answer and interpret what this value represents in terms of the lake's environmental remediation.

  1. The temperature of a metal bar of length 3 meters depends on the distance x from its left end. The temperature is modeled by the function T(x) = 40 + 15x² degrees Celsius for 0 ≤ x ≤ 3. Determine the average temperature of the entire metal bar.
    Average Temperature:

  2. The acceleration of water flow rate into a reservoir is modeled by f''(t) = 6t - 4 gallons per minute squared, where t is measured in minutes. At time t = 0, the initial rate of water flow into the reservoir is f'(0) = 10 gallons per minute, and the initial volume of water stored in the reservoir is f(0) = 100 gallons.

Part A: Find the equation for the water flow rate f'(t) at any time t.
f'(t) =

Part B: Find the equation for the volume of water f(t) in the reservoir at any time t.
f(t) =

Answers and Explanations

  1. D. 64 meters
    Explanation: The velocity function is v(t) = 3t² - 12t = 3t(t - 4). The velocity is negative on [0, 4] and positive on [4, 5]. The net change is the integral from 0 to 5 of v(t) dt, which is [t³ - 6t²] evaluated from 0 to 5, giving 125 - 150 = -25. The total distance is the integral of |v(t)| dt. From 0 to 4, the integral of -v(t) dt is -[t³ - 6t²] from 0 to 4, which is -(64 - 96) = 32. From 4 to 5, the integral of v(t) dt is [t³ - 6t²] from 4 to 5, which is -25 - (-32) = 7. Thus, total distance is 32 + 7 = 39. The difference between total distance and net change is 39 - (-25) = 64.

  2. B. 0.5 - 0.5 × exp(-2)
    Explanation: The average value of f(x) over [0, 4] is (1 / (4 - 0)) × integral from 0 to 4 of exp(-0.5x) dx. The antiderivative of exp(-0.5x) is -2 × exp(-0.5x). Evaluating from 0 to 4 gives -2 × exp(-2) - (-2 × exp(0)) = 2 - 2 × exp(-2). Multiplying by 1/4 gives 0.5 - 0.5 × exp(-2).

  3. C. The average rate of water leakage in liters per hour between 2:00 AM and 8:00 AM.
    Explanation: The expression (1 / (8 - 2)) × integral from 2 to 8 of C(t) dt represents the average value of the rate of change function C(t) over the interval [2, 8], which translates to the average rate of water leakage in liters per hour between 2:00 AM and 8:00 AM.

  4. B. 144 gallons
    Explanation: The interval [0, 12] is divided into four subintervals of length delta_t = 3: [0, 3], [3, 6], [6, 9], and [9, 12]. The Right Riemann Sum is 3 × (r(3) + r(6) + r(9) + r(12)) = 3 × (13 + 15 + 12 + 8) = 3 × 48 = 144.

  5. B. 122 gallons
    Explanation: The net amount of water at t = 6 is given by W(6) = W(0) + integral from 0 to 6 of (E(t) - L(t)) dt = 50 + integral from 0 to 6 of (8t - t²) dt. The antiderivative is 4t² - t³/3. Evaluating from 0 to 6 gives [4(36) - 216/3] = 144 - 72 = 72. Adding the initial 50 gallons gives 50 + 72 = 122 gallons.

  6. A. -2√(π)
    Explanation: Using the Leibniz Rule (Fundamental Theorem of Calculus Part 1), g'(x) = cos(x²) × d/dx(x²) = 2x × cos(x²). Evaluating at x = √(pi) gives g'(√(pi)) = 2 × √(pi) × cos(pi) = 2 × √(pi) × (-1) = -2 × √(pi).

  7. C. 1240 bees
    Explanation: The maximum population occurs where P'(t) = 0 or at the endpoints. P'(t) = 120 - 30t = 0 when t = 4 weeks. Since P'(t) > 0 for t < 4 and P'(t) < 0 for t > 4, a global maximum occurs at t = 4. The population is P(4) = P(0) + integral from 0 to 4 of (120 - 30t) dt = 1000 + [120t - 15t²] from 0 to 4 = 1000 + (480 - 240) = 1240 bees.

  8. A. 2
    Explanation: The definite integral of f(x) from 0 to 8 is given by the areas above the x-axis minus the area below the x-axis: integral from 0 to 8 of f(x) dx = 12 - 4 + 8 = 16. The average value is (1 / (8 - 0)) × integral from 0 to 8 of f(x) dx = 16 / 8 = 2.

  9. Part A: -4 units, Part B: 12 units
    Explanation: The function f'(t) consists of triangles. Segment A from (0,4) to (2,0) forms a right triangle above the axis with area 0.5 × 2 × 4 = 4. Segment B and C from (2,0) to (6,0) with a vertex at (4,-4) form a triangle below the axis with area 0.5 × 4 × 4 = 8. Part A: Net change is 4 - 8 = -4. Part B: Total accumulated change is 4 + 8 = 12.

  10. Part A: 0, Part B: 2/π meters per second
    Explanation: Part A: Average velocity is (1/2) × integral from 0 to 2 of cos(pi × t) dt = (1/2) × [sin(pi × t) / pi] evaluated from 0 to 2 = (1/(2 × pi)) × (sin(2 × pi) - sin(0)) = 0. Part B: Speed is |v(t)|. The integral from 0 to 2 of |cos(pi × t)| dt is four times the integral from 0 to 0.5 of cos(pi × t) dt, which is 4 × [sin(pi × t) / pi] from 0 to 0.5 = 4/pi. Dividing by 2 for the interval length gives (4/pi)/2 = 2/pi.

  11. Part A: H(t) = 20√(t³) - 5t² + 150, Part B: 230 meters
    Explanation: Part A: Integrating H'(t) = 30 × √(t) - 10t yields H(t) = 20 × √(t³) - 5 × t² + C. Using H(0) = 150, we find C = 150, so H(t) = 20 × √(t³) - 5 × t² + 150. Part B: At t = 4, H(4) = 20 × √(4³) - 5 × (4)² + 150 = 20 × 8 - 80 + 150 = 160 - 80 + 150 = 230 meters.

  12. Part A: 316.060, Part B: Units are kilograms. This represents the total mass of the chemical pollutant removed from the lake over the first 10 days of filtration.
    Explanation: Part A: The total mass removed is the integral from 0 to 10 of 50 × exp(-0.1t) dt. The antiderivative is -500 × exp(-0.1t). Evaluating from 0 to 10 gives -500 × exp(-1) - (-500) = 500 × (1 - exp(-1)) approx 500 × (1 - 0.3678794) = 316.06028... which rounds to 316.060 kilograms. Part B: The units are kilograms, and it represents the total mass of pollutant removed.

  13. 85 degrees Celsius
    Explanation: The average temperature is (1/3) × integral from 0 to 3 of (40 + 15x²) dx. The antiderivative is 40x + 5x³. Evaluating from 0 to 3 yields 40(3) + 5(27) = 120 + 135 = 255. Dividing by 3 gives 255 / 3 = 85.

  14. Part A: f'(t) = 3t² - 4t + 10, Part B: f(t) = t³ - 2t² + 10t + 100
    Explanation: Part A: Integrating f''(t) = 6t - 4 gives f'(t) = 3t² - 4t + C. Using f'(0) = 10, we get C = 10, so f'(t) = 3t² - 4t + 10. Part B: Integrating f'(t) gives f(t) = t³ - 2t² + 10t + K. Using f(0) = 100, we get K = 100, so f(t) = t³ - 2t² + 10t + 100.