AP Calculus BC Flashcards: Applying Properties of Definite Integrals
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
How does the property for the integral of a constant times a function work?
The constant can be factored out of the integral. For example, ∫ₐᵇ c*f(x) dx is equal to c * ∫ₐᵇ f(x) dx.
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How does the property for the integral of a constant times a function work?
The constant can be factored out of the integral. For example, ∫ₐᵇ c*f(x) dx is equal to c * ∫ₐᵇ f(x) dx.
List four fundamental properties of definite integrals.
Properties include handling a constant times a function, the sum of two functions, the reversal of limits of integration, and combining integrals over adjacent intervals.
What is the 'adjacent intervals' property of definite integrals?
This property states that the integral of a function over an interval [a, c] is the sum of the integrals over adjacent subintervals, such as ∫ₐᶜ f(x) dx = ∫ₐᵇ f(x) dx + ∫ᵦᶜ f(x) dx.
What is a method for evaluating a definite integral that relies on its graphical interpretation?
A definite integral can be evaluated by using geometry and the connection between the definite integral and the net area under the function's curve.
What two specific types of discontinuities can a function have and still be integrable?
A function can still be integrable if it has a finite number of removable or jump discontinuities over the interval of integration.
How is the integral of a sum of two functions, ∫ₐᵇ [f(x) + g(x)] dx, calculated using integral properties?
The integral of a sum is the sum of the integrals, so it can be separated into two integrals: ∫ₐᵇ f(x) dx + ∫ₐᵇ g(x) dx.
What is the effect of reversing the limits of integration on a definite integral?
Reversing the limits of integration negates the value of the definite integral; that is, ∫ₐᵇ f(x) dx = -∫ᵦᵃ f(x) dx.
Can definite integrals be evaluated for functions that are not continuous over the interval?
Yes, the definition of the definite integral can be extended to functions that have a finite number of removable or jump discontinuities.
What are two distinct approaches you can use to calculate a definite integral, based on the provided content?
You can calculate a definite integral by using its connection to geometric areas or by applying the algebraic properties of definite integrals (like sum, constant multiple, etc.).
If the graph of f(x) from x=a to x=b is a semicircle above the x-axis, how would you calculate the definite integral ∫ₐᵇ f(x) dx?
You would calculate the definite integral by finding the area of the corresponding geometric shape, in this case, the area of the semicircle.