AP®︎ Calculus AB: Unit 6 Practice Test
Prepare for your quiz, test, or the AP exam with focused practice questions on Unit 6 of AP Calculus AB – Integration and Accumulation of Change.
Questions List
Unit 6 (All Topics)
Question 1 Topic 6.1Easy
A pump moves water into a pool at a constant rate of 12 gallons per minute for 8 minutes. How many gallons of water enter the pool during those 8 minutes?
What You’re Being Tested On:
Explore the learning objectives taken directly from the College Board’s AP® Calculus AB Curriculum. Ensure you’re prepared for the exact topics covered on the AP® exam, in-class tests, and quizzes, and gain confidence in your mastery of the material.
Unit 6: Integration and Accumulation of Change
Learn definite integrals, the Fundamental Theorem of Calculus, Riemann sums, and antiderivatives.
Topic 6.1: Exploring Accumulations of Change
Learning Objective: CHA-4.A
Interpret the meaning of areas associated with the graph of a rate of change in context.
Essential Knowledge: CHA-4.A.1
The area of the region between the graph of a rate of change function and the x axis gives the accumulation of change.
Essential Knowledge: CHA-4.A.2
In some cases, accumulation of change can be evaluated by using geometry.
Essential Knowledge: CHA-4.A.3
If a rate of change is positive (negative) over an interval, then the accumulated change is positive (negative).
Essential Knowledge: CHA-4.A.4
The unit for the area of a region defined by rate of change is the unit for the rate of change multiplied by the unit for the independent variable.
Topic 6.2: Approximating Areas with Riemann Sums
Learning Objective: LIM-5.A
Approximate a definite integral using geometric and numerical methods.
Essential Knowledge: LIM-5.A.1
Definite integrals can be approximated for functions that are represented graphically, numerically, analytically, and verbally.
Essential Knowledge: LIM-5.A.2
Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or nonuniform partitions.
Essential Knowledge: LIM-5.A.3
Definite integrals can be approximated using numerical methods, with or without technology.
Essential Knowledge: LIM-5.A.4
Depending on the behavior of a function, it may be possible to determine whether an approximation for a definite integral is an underestimate or overestimate for the value of the definite integral.
Topic 6.3: Riemann Sums, Summation Notation, and Definite Integral Notation
Learning Objective: LIM-5.B
Interpret the limiting case of the Riemann sum as a definite integral.
Essential Knowledge: LIM-5.B.1
The limit of an approximating Riemann sum can be interpreted as a definite integral.
Essential Knowledge: LIM-5.B.2
A Riemann sum, which requires a partition of an interval I, is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition.
Learning Objective: LIM-5.C
Represent the limiting case of the Riemann sum as a definite integral.
Essential Knowledge: LIM-5.C.1
The definite integral of a continuous function f over the interval [a, b], denoted by ∫_a^b f(x) dx, is the limit of Riemann sums as the widths of the subintervals approach 0. That is, ∫_a^b f(x) dx = lim_{max Δx_i → 0} Σ_{i=1}^n f(x_i^*) Δx_i, where n is the number of subintervals, Δx_i is the width of the i th subinterval, and x_i^* is a value in the i th subinterval.
Essential Knowledge: LIM-5.C.2
A definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.
Topic 6.4: The Fundamental Theorem of Calculus and Accumulation Functions
Learning Objective: FUN-5.A
Represent accumulation functions using definite integrals.
Essential Knowledge: FUN-5.A.1
The definite integral can be used to define new functions. Illustrative example: f(x) = ∫_0^x e^t dt = e^x - 1.
Essential Knowledge: FUN-5.A.2
If f is a continuous function on an interval containing a, then d/dx ∫_a^x f(t) dt = f(x), where x is in the interval.
Topic 6.5: Interpreting the Behavior of Accumulation Functions Involving Area
Learning Objective: FUN-5.A
Represent accumulation functions using definite integrals.
Essential Knowledge: FUN-5.A.3
Graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as g(x) = \u222b_a^x f(t) dt.
Topic 6.6: Applying Properties of Definite Integrals
Learning Objective: FUN-6.A
Calculate a definite integral using areas and properties of definite integrals.
Essential Knowledge: FUN-6.A.1
In some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area.
Essential Knowledge: FUN-6.A.2
Properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals.
Essential Knowledge: FUN-6.A.3
The definition of the definite integral may be extended to functions with removable or jump discontinuities.
Topic 6.7: The Fundamental Theorem of Calculus and Definite Integrals
Learning Objective: FUN-6.B
Evaluate definite integrals analytically using the Fundamental Theorem of Calculus.
Essential Knowledge: FUN-6.B.1
An antiderivative of a function f is a function g whose derivative is f.
Essential Knowledge: FUN-6.B.2
If a function f is continuous on an interval containing a, the function defined by F(x) = ∫_a^x f(t) dt is an antiderivative of f for x in the interval.
Essential Knowledge: FUN-6.B.3
If f is continuous on the interval [a, b] and F is an antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a).
Topic 6.8: Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
Learning Objective: FUN-6.C
Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.
Essential Knowledge: FUN-6.C.1
∫ f(x) dx is an indefinite integral of the function f and can be expressed as ∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is any constant.
Essential Knowledge: FUN-6.C.2
Differentiation rules provide the foundation for finding antiderivatives.
Essential Knowledge: FUN-6.C.3
Many functions do not have closed-form antiderivatives.
Topic 6.9: Integrating Using Substitution
Learning Objective: FUN-6.D
For integrands requiring substitution or rearrangements into equivalent forms: (a) Determine indefinite integrals. (b) Evaluate definite integrals.
Essential Knowledge: FUN-6.D.1
Substitution of variables is a technique for finding antiderivatives.
Essential Knowledge: FUN-6.D.2
For a definite integral, substitution of variables requires corresponding changes to the limits of integration.
Topic 6.10: Integrating Functions Using Long Division and Completing the Square
Learning Objective: FUN-6.D
For integrands requiring substitution or rearrangements into equivalent forms: (a) Determine indefinite integrals. (b) Evaluate definite integrals.
Essential Knowledge: FUN-6.D.3
Techniques for finding antiderivatives include rearrangements into equivalent forms, such as long division and completing the square.
Topic 6.14: Selecting Techniques for Antidifferentiation
