AP Calculus BC Flashcards: The Fundamental Theorem of Calculus and 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.
If $F$ is an antiderivative of a continuous function $f$, what does the value $F(b) - F(a)$ represent?
The value $F(b) - F(a)$ represents the definite integral of $f$ from $a$ to $b$, which is written as $\int_{a}^{b} f(x) dx$.
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If $F$ is an antiderivative of a continuous function $f$, what does the value $F(b) - F(a)$ represent?
The value $F(b) - F(a)$ represents the definite integral of $f$ from $a$ to $b$, which is written as $\int_{a}^{b} f(x) dx$.
What is the relationship between the derivative of a function $g$ and the function $f$ if $g$ is an antiderivative of $f$?
The derivative of the function $g$ is the function $f$.
If $F(x) = \int_{5}^{x} \sin(t) dt$, what is $F'(x)$?
Based on the Fundamental Theorem of Calculus, the derivative of this integral function is the original integrand with the variable $x$, so $F'(x) = \sin(x)$.
Use the Fundamental Theorem of Calculus to evaluate $\int_{0}^{2} 4x^3 dx$.
An antiderivative of $4x^3$ is $F(x) = x^4$. The integral is $F(2) - F(0) = 2^4 - 0^4 = 16$.
State the part of the Fundamental Theorem of Calculus used for evaluating definite integrals (also known as the Evaluation Theorem).
If $f$ is continuous on the interval $[a, b]$ and $F$ is an antiderivative of $f$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$.
How do you analytically evaluate the definite integral $\int_{a}^{b} f(x) dx$?
First, find an antiderivative $F$ of $f$. Then, evaluate the expression $F(b) - F(a)$ to find the value of the integral.
What condition must a function $f$ meet on an interval $[a, b]$ to apply the evaluation part of the Fundamental Theorem of Calculus?
The function $f$ must be continuous on the interval $[a, b]$.
State the part of the Fundamental Theorem of Calculus that defines an antiderivative using a definite integral.
If a function $f$ is continuous on an interval containing $a$, the function defined by $F(x) = \int_{a}^{x} f(t) dt$ is an antiderivative of $f$.
What is an antiderivative of a function $f$?
An antiderivative of a function $f$ is a function $g$ whose derivative is $f$.
What is the derivative of the function $F(x) = \int_{a}^{x} f(t) dt$?
The derivative of $F(x)$ is $f(x)$, because the function $F(x)$ is defined as an antiderivative of $f$.