AP®︎ Calculus BC: Unit 6 Practice Test
Prepare for your quiz, test, or the AP exam with focused practice questions on Unit 6 of AP Calculus BC – Integration and Accumulation of Change.
Questions List
Unit 6 (All Topics)
Question 1 Topic 6.1Easy
A pump drains a swimming pool at a constant rate of 18 gallons per minute for 25 minutes.
What is the change in the volume of water in the pool over these 25 minutes?
What You’re Being Tested On:
Explore the learning objectives taken directly from the College Board’s AP® Calculus BC 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, FTC, Riemann sums, antiderivatives, and improper integrals.
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 a 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 ith subinterval, and x_i^* is a value in the ith 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 examples: f(x) = \int_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 \int_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) = ∫_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 6.11: Integrating Using Integration by Parts
Learning Objective: FUN-6.E
For integrands requiring integration by parts: (a) Determine indefinite integrals. (b) Evaluate definite integrals.
Essential Knowledge: FUN-6.E.1
Integration by parts is a technique for finding antiderivatives.
Topic 6.12: Integrating Using Linear Partial Fractions
Learning Objective: FUN-6.F
For integrands requiring integration by linear partial fractions: (a) Determine indefinite integrals. (b) Evaluate definite integrals.
Essential Knowledge: FUN-6.F.1
Some rational functions can be decomposed into sums of ratios of linear, nonrepeating factors to which basic integration techniques can be applied.
Topic 6.13: Evaluating Improper Integrals
Learning Objective: LIM-6.A
Evaluate an improper integral or determine that the integral diverges.
Essential Knowledge: LIM-6.A.1
An improper integral is an integral that has one or both limits infinite or has an integrand that is unbounded in the interval of integration.
Essential Knowledge: LIM-6.A.2
Improper integrals can be determined using limits of definite integrals.
Topic 6.14: Selecting Techniques for Antidifferentiation
