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AP Calculus AB Unit 6 Review and Practice Test

Integration and Accumulation of Change

The AP Calculus AB Unit 6 review covers integration and accumulation of change, key ideas that link rates, areas, and motion. Build your skills with Unit 6 AP Calc AB progress check MCQ Part A and FRQ Part A practice to strengthen your understanding.

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Get Exam-Ready with Our AP Calculus AB Unit 6 Review

The AP Calc AB Unit 6 progress check helps you review integration and accumulation through targeted practice. Work on MCQ Part A and Part B, FRQ Part A and Part B, and real-world rate in–rate out FRQs to refine your understanding. This Unit 6 AP Calculus AB review also includes questions on integration and accumulation of change, letting you master everything from definite integrals to total area and net change problems. Use the AP Calc AB Unit 6 practice test to evaluate your progress and prepare confidently for the Unit 6 AP Calculus AB test.

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Learn with AP Calculus AB Unit 6 video lessons that simplify complex ideas like integration and accumulation of change. Each session breaks down Unit 6 progress check MCQs and FRQ problems step by step, helping you understand how to apply formulas to motion, area, and rate scenarios. These AP Calculus AB integration practice videos guide you through real exam-style questions, so you can build both confidence and accuracy before your Unit 6 AP Calc AB test.

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Explore interactive AP Calculus AB Unit 6 study guides that break down tough topics like integration, accumulation, and rate of change in simple terms. Each guide includes summaries, solved examples, and quick tips for tackling Unit 6 progress check FRQ Part B and MCQ questions. Use these AP Calculus AB Unit 6 review materials to reinforce your understanding and prepare efficiently for the AP Calculus AB Unit 6 test.

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Sharpen Your Skills with AP Calc AB Unit 6 Practice Tests

Boost your understanding with AP Calc AB Unit 6 practice tests focused on integration, accumulation, and rate of change. Work through Unit 6 progress check MCQ Part A and Part B, plus FRQ practice questions, to simulate real exam conditions. These AP Calculus AB Unit 6 review tests help you apply concepts, improve speed, and gain confidence for your AP Calculus AB Unit 6 test
Try these sample practice questions with detailed answer explanations:
Integration and Accumulation of Change Practice Tests

Question

The graph of a differentiable function f is shown above. If g (x) = ∫ 2 x f (t) dt , which of the following is true?

A. g(5) < g′(5) < g″(5)
B. g(5) < g″(5) < g′(5)
C. g′(5) < g(5) < g″(5)
D. g″(5) < g′(5) < g(5)

Hint :

Determine whether each quantity is negative, zero, or positive and then compare their relative values.

Explanation

To compare the values of g(5), g′(5), and g″(5), evaluate g(5) and find the first and second derivatives of g(x) at x = 5.

Evaluate g(5)

Plug x = 5 into g(x) to evaluate g(5): g (5) = ∫ 2 5 f (t) dt

This definite integral represents the area bounded by the graph of f and the x-axis between x = 2 and x = 5. The exact value of g(5) cannot be determined, but this region is above the x-axis, so g(5) > 0.

Evaluate g′(5)

The given function is of the form ∫ a x f (t) dt , so use the Second Fundamental Theorem of Calculus (Second FTC) to differentiate g(x) and then plug in x = 5.

g (x) = ∫ 2 x f (t) dt Given function
g ′ (x) = d dx [ ∫ 2 x f (t) dt ] Differentiate both sides
g ′ (x) = f (x) Apply Second FTC
g ′ (5) = f (5) Plug in x = 5

Examine the graph of f at x = 5 to see that g′(5) = 0.

Evaluate g″(5)

Differentiate g′(x) = f(x) and then plug in x = 5 to evaluate g″(5).

g ′ (x) = f (x) First derivative of g (x)
d dx [ g ′ (x) ] = d dx [ f (x) ] Differentiate both sides
g ″ (x) = f ′ (x) Simplify
g ″ (5) = f ′ (5) Plug in x = 5

The value of f ′ (5) is the slope of the line tangent to f at x = 5. The exact value of g″(5) cannot be found, but the slope of the tangent line is negative at x = 5, so g″(5) < 0.

Therefore, the statement g″(5) < g′(5) < g(5) is true because g″(5) < 0, g′(5) = 0, and g(5) > 0.

(Choice A) g(5) < g′(5) < g″(5) may result from mistakenly reversing the correct inequality.

(Choice B) g(5) < g″(5) < g′(5) may result from incorrectly calculating the relative values of the three quantities.

(Choice C) g′(5) < g(5) < g″(5) may result from assuming the graph represents g(x) rather than f(x).

Things to remember:
The derivative of an integral of the form ∫ a x f (t) dt , where f is a continuous function, is f(x):

d dx [ ∫ a x f (t) dt ] = f (x)

Question

The rate at which gravel is deposited at a landfill on a certain day is modeled by the function R, where R(t) is measured in tons per hour and t is the number of hours since the landfill opened. Using a trapezoidal sum with three subintervals as indicated by the data in the table above, approximate the number of tons of gravel deposited during the first 8 hours since the landfill opened.

A. 75
B. 81
C. 128
D. 150

Hint :

The net change in a quantity over an interval of time is equal to the definite integral of the rate of change in the quantity over that interval.

Explanation

The number of tons of gravel deposited during the first 8 hours since the landfill opened is the net change in tons of gravel deposited at the rate R(t) from t1 = 0 to t2 = 8.

The net change in a quantity over an interval of time [t1, t2] is equal to the definite integral of the rate of change r(t) in the quantity over that interval.

Therefore, the number of tons of gravel deposited during the first 8 hours since the landfill opened is equal to ∫ 0 8 R (t) dt .

To approximate the net change integral using a trapezoidal sum with three subintervals, calculate the areas of three trapezoids, where the trapezoid on subinterval n has height hn and base lengths bn and bn+1.

The height hn of each trapezoid is the length of the subinterval (distance between the t-values). The base lengths bn and bn+1 of each trapezoid are the values of R(t) at the endpoints of the subinterval.

The three subintervals from the given table are [0, 2], [2, 5], and [5, 8], so the heights of the trapezoids are 2, 3, and 3, respectively. The base lengths are the given values of R at the endpoints.

Plug values of bn and h for the three trapezoids into the trapezoidal sum expression and simplify. Factor the greatest common factor (GCF)  1 2  out of the trapezoidal sum expression for a more efficient calculation.

1 2 h 1 (b 1 + b 2 ) + 1 2 h 2 (b 2 + b 3 ) + 1 2 h 3 (b 3 + b 4 ) Trapezoidal sum
1 2 [h 1 (b 1 + b 2 ) + h 2 (b 2 + b 3 ) + h 3 (b 3 + b 4 ) ] . Factor out GCF  1 2
1 2 [(2) (4 + 8) + (3) (8 + 10) + (3) (10 + 14)] Plug in values of bn and hn
1 2 [24 + 54 + 72] Simplify inside brackets
75 Simplify

Therefore, the approximate number of tons of gravel deposited during the first 8 hours since the landfill opened is 75.

(Choice B) 81 may result from mistakenly assuming that each of the three trapezoids has the same height (3).

(Choice C) 128 may result from approximating the number of tons of gravel using a left Riemann sum, but the question asks for the approximation using a trapezoidal sum.

(Choice D) 150 may result from neglecting the factors of  1 2  in the trapezoidal sum equation.

Things to remember:

  • The net change in a quantity over an interval of time [t1, t2] is given by the definite integral of the rate of change r(t) in the quantity from t1 to t2.
  • A trapezoidal sum approximates a definite integral  ∫ a b f (x) dx  using trapezoids with height hn and base lengths bn and bn+1 to estimate the area under f in each subinterval.

∫ a b f (x) dx ≈ 1 2 h 1 (b 1 + b 2) + 1 2 h 2 (b 2 + b 3) + …

Question

∫ 2 x 3 (x 4 - 8) 5 d x =

A. 1 2 x 4 ( 1 5 x 5 - 8 x ) 5 + C
B. 1 12 ( x 4 - 8 ) 6 + C
C. 1 3 ( x 4 - 8 ) 6 + C
D. 4 3 ( x 4 - 8 ) 6 + C

Hint :

If one factor of an integrand is a constant multiple of the derivative of the inside of a composite function, use  u-substitution to integrate.

Explanation

If one term of the integrand is a constant multiple of the derivative of the inside of a composite function, use u -substitution to integrate.

.

The given integrand contains the composite function  (x 4 - 8) 5 , where  u = x 4 - 8  is the inside function.

Check to see if the factor 2x3 from the integrand is a constant multiple of the derivative of x4−8 .

Since  2 x 3  is a constant multiple of  4 x 3 , use u-substitution to rewrite the integral in terms of u, where u=x4−8 , First differentiate u with respect to x, and then solve for dx.

u = x 4 - 8 Inside function
d u d x = 4 x 3 Differentiate both sides with respect to  x
d u = 4 x 3 d x Multiply both sides by  d x
d u 4 x 3 = d x Divide both sides by  4 x 3  to solve for  d x

Now substitute u for x 4 - 8  and d u 4 x 3 for dx into the given integral, and then simplify and integrate.

∫ 2 x 3 (x 4 - 8) 5 d x Given integral
∫ 2 x 3 (u) 5 d u 4 x 3 Substitute  u = x 4 - 8  and  d u 4 x 3 = d x
2 4 ∫ x 3 (u) 5 d u x 3 Reduce:  x 3 x 3 = 1  and apply:  ∫ c f (x) d x = c ∫ f (x) d x
1 2 ∫ u 5 d u Reduce:  2 4 = 1 2
1 2 ⋅ u 5 + 1 6 + C Apply:  ∫ x n   d x = x n + 1 n + 1 + C
u 6 12 + C Simplify

Substitute x 4 - 8 back in for u to write the answer in terms of x .

u 6 12 + C Answer in terms of  u
(x 4 - 8) 6 12 + C Substitute  x 4 - 8  back in for  u
1 12 (x 4 - 8) 6 + C Rewrite to match answer choice

Therefore,  ∫ 2 x 3 (x 4 - 8) 5 d x = 1 12 (x 4 - 8) 6 + C .

(Choice A) 1 2 x 4 ( 1 5 x 5 - 8 x ) 5 + C  is a result of integrating each factor independently instead of using  u -substitution.

(Choice C) 1 3 (x 4 - 8) 6 + C  may be a result of omitting  1 4  when substituting in  d u 4 x 3  for  d x .

(Choice D) 4 3 (x 4 - 8) 6 + C  may be a result of mistakenly rewriting  d u 4 x 3  as  4 d u x 3 .

Things to remember:
If one term of the integrand is a constant multiple of the derivative of the inside function of a composite function, use u-substitution to integrate.

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Frequently Asked Questions (FAQs)

AP Calculus AB Unit 6 focuses on understanding integration as a tool for measuring accumulation, area, and total change, one of the most essential parts of the course and the AP exam. Here’s what you’ll explore in this unit:

  • Using definite integrals to determine accumulated change over an interval: Learn how to calculate total distance, area, or change by evaluating definite integrals. You’ll see how these apply to rate-in, rate-out FRQs and real-world motion problems.
  • Approximating integrals using Riemann Sums: Understand how Riemann sums estimate the value of an integral by dividing curves into rectangles or trapezoids. These questions often appear in the AP Calculus AB Unit 6 Progress Check MCQs.
  • Accumulation functions, the Fundamental Theorem of Calculus, and definite integrals: Connect differentiation and integration through the Fundamental Theorem of Calculus and use accumulation functions to describe change over time.
  • Antiderivatives and indefinite integrals: Practice finding the original function when given its derivative, a core skill for solving both MCQs and FRQs in this unit.
  • Properties of integrals and integration techniques: Master key rules like linearity and the difference of functions, along with substitution and symmetry strategies to simplify complex integrals.

Every one of these concepts builds toward fluency with the Integration and Accumulation of Change unit, which carries significant weight on the AP Calculus AB exam. For targeted explanations and realistic practice, explore UWorld’s detailed video lessons and step-by-step solutions tailored for Unit 6 success.

To prepare effectively for the AP Calc AB Unit 6 exam, start with a structured plan that covers both conceptual understanding and practice. Review all topics on integration and accumulation of change, focusing on definite integrals, accumulation models, and applications of rate functions.

Here’s how to get started:

  • Work through unit 6 progress check MCQ Part A and Part B Calc AB sets regularly.
  • Practice FRQs from both unit 6 progress check Part A and Part B of AP Calculus AB to build exam endurance.
  • Keep a calc AB Unit 6 cheat sheet or review summary handy for formulas and key steps.
  • Time your sessions to simulate test-day conditions.

When you’re ready to take your prep further, UWorld’s AP Calculus AB integration test questions and detailed explanations will help you master every topic, making sure you go into the exam confident and prepared.

Absolutely. For free AP Calculus AB Unit 6 integration and accumulation resources, start with the College Board’s AP Classroom, which includes Unit 6 progress check MCQs and FRQs for both Calc AB and Calculus AB. Khan Academy also offers free video lessons explaining integration concepts and accumulation models in clear detail.

However, while free tools are great for concept reinforcement, they don’t always reflect true AP-level difficulty. For more rigorous prep, UWorld’s AP Calc AB Unit 6 integration practice offers exam-style questions with step-by-step solutions that help you identify and strengthen weak areas. You’ll also receive instant feedback and performance analytics to guide your study.

The best part? You can try UWorld free for 7 days to experience realistic AP-level practice questions and detailed explanations. Combining free resources for review with UWorld’s 7-day trial gives you the perfect balance of concept clarity and exam mastery for the Unit 6 AP Calculus AB test.

The AP Calculus AB Unit 6 test includes a mix of multiple-choice (MCQ) and free-response (FRQ) questions that test your understanding of integration and its applications. Expect:

  • MCQs on antiderivatives, definite integrals, and area under curves.
  • FRQs require explanations of rate in–rate out and net accumulation.
  • Modeling problems that apply integration to motion, population, or growth.

You’ll practice problems similar to those in the AP Calc AB Unit 6 Progress Check MCQ Part A and B and the FRQ Part A and B, helping you get comfortable with the question formats used on the AP Calculus AB exam.

UWorld’s calc AB Unit 6 test practice helps you become familiar with the pacing, structure, and logic behind each question type. With repeated practice, you’ll not only improve accuracy but also develop the problem-solving confidence needed to ace the AP Calculus AB Unit 6 test.

Improving your FRQ performance in AP Calculus AB Unit 6 starts with mastering the method behind each problem type. The key is to understand the relationship between integration and accumulation of change and how to clearly communicate your reasoning.

Here’s how to strengthen your approach:

  • Practice rate accumulation AP Calc AB FRQs to build interpretation skills.
  • Review unit 6 progress check FRQ Part B Calc AB for multi-step integration setups.
  • Focus on proper notation, justification, and explanation, all essential for scoring full credit.

UWorld’s AP Calculus AB Unit 6 FRQ practice includes detailed sample responses and scoring notes that mirror College Board standards. With regular practice, you’ll learn how to organize your work, avoid common mistakes, and earn top marks on this high-value section of the test.

The Integration and Accumulation of Change unit makes up roughly 17–20% of the AP Calculus AB exam, one of the most heavily tested sections. Because it connects directly to concepts from Units 5, 7, and 8, mastering Unit 6 gives you an edge across multiple parts of the test.

Students often find that Unit 6 progress check MCQ AP Calculus AB questions overlap with later unit FRQs, reinforcing key calculus principles. Focusing your preparation here ensures that you’re covering a major portion of the scoring weight.

UWorld’s integration and accumulation practice tests help you focus on these high-yield questions with step-by-step explanations that turn complex problems into understandable, repeatable strategies; giving you the confidence to score higher.

An effective AP Calculus AB Unit 6 study guide should summarize integration techniques, include accumulation examples, and feature plenty of practice problems. Look for guides that explain concepts like average value, total change, and area between curves using clear visuals and problem walkthroughs.

UWorld’s AP Calc AB Unit 6 review PDF offers all of this and more. It combines concise explanations, solved examples, and progress-check exercises that mirror exam-style questions. The Unit 6 Calculus AB review materials make studying efficient, whether you’re brushing up on fundamentals or preparing for full-length practice tests.

Best of all, you can try UWorld’s free trial for AP Calculus AB course before upgrading for full access. Experience the difference as you master integration and accumulation of change concepts with realistic practice and in-depth explanations.

Yes! You can find several AP Calculus AB Unit 6 practice tests that focus on integration and accumulation concepts. These include the Unit 6 Progress Check MCQ Part A and Part B and FRQ Part A and Part B for AP Calc AB. You can also explore Unit 6 progress check MCQ and FRQ practice with answers to review detailed solutions and strengthen your understanding before test day.

  • Practicing these tests helps you:
  • Build familiarity with AP-style problem formats.
  • Identify weak topics like net change or rate in–rate out problems.
  • Improve your speed and accuracy under time pressure.

UWorld offers tailored Unit 6 AP Calculus AB practice sessions with in-depth feedback, performance tracking, and timed quizzes that simulate the real AP exam. It’s the most effective way to prepare thoroughly for the AP Calculus AB Unit 6 test and walk into exam day with confidence.

Learn More About Specific Unit

Strengthen your grasp of AP Calculus AB by exploring related units that build on these concepts.
Unit 1

Limits and Continuity

Unit 2

Differentiation: Definition and Fundamental Properties
Unit 3

Differentiation: Composite, Implicit, and Inverse Functions
Unit 4

Contextual Applications of Differentiation
Unit 5

Analytical Applications of Differentiation
Unit 7

Differential Equations
Unit 8

Applications of Integration
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