AP Calculus AB 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.
State the evaluation part of the Fundamental Theorem of Calculus.
If a function $f$ is continuous on the interval $[a, b]$ and $F$ is an antiderivative of $f$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$.
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State the evaluation part of the Fundamental Theorem of Calculus.
If a function $f$ is continuous on the interval $[a, b]$ and $F$ is an antiderivative of $f$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$.
What is the derivative of the function $G(x) = \int_{a}^{x} f(t) dt$ with respect to $x$?
The derivative of $G(x)$ is $f(x)$, as the function defined by the integral 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 primary analytical method for evaluating definite integrals?
Definite integrals are evaluated analytically by finding an antiderivative of the integrand and then applying the Fundamental Theorem of Calculus.
If $F$ is an antiderivative of $f$, what is the final step in calculating $\int_{a}^{b} f(x) dx$?
The final step is to calculate the value of the antiderivative at the upper and lower limits and find their difference, $F(b) - F(a)$.
How can an integral be used to define an antiderivative of a function $f$?
If $f$ 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 the relationship between differentiation and integration as described by the two parts of the Fundamental Theorem of Calculus?
The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse operations.
Given that $g'(x) = f(x)$, express the value of $\int_{2}^{7} f(x) dx$ in terms of the function $g$.
Since $g$ is an antiderivative of $f$, the definite integral $\int_{2}^{7} f(x) dx$ is equal to $g(7) - g(2)$ by the Fundamental Theorem of Calculus.
What does the expression $F(b) - F(a)$ represent in the Fundamental Theorem of Calculus?
It represents the net change of the antiderivative $F$ over the interval $[a, b]$, which is equivalent to the value of the definite integral of its derivative, $f$.
What condition must a function $f$ meet to apply the evaluation part of the Fundamental Theorem of Calculus on $[a, b]$?
The function $f$ must be continuous on the closed interval $[a, b]$.