Evaluation theorem

Definition

The Evaluation Theorem states that the integral of a continuous function over an interval can be found using its antiderivative. Specifically, if $F$ is an antiderivative of $f$, then $\int_a^b f(x) \, dx = F(b) - F(a)$.

5 Must Know Facts For Your Next Test

1. The Evaluation Theorem is a direct consequence of the Fundamental Theorem of Calculus.
2. It requires finding an antiderivative $F$ such that $F'(x) = f(x)$.
3. The theorem simplifies the process of calculating definite integrals.
4. Both the lower limit $a$ and the upper limit $b$ are essential in evaluating the integral.
5. The notation $\int_a^b f(x) \, dx$ represents the area under the curve from $x=a$ to $x=b$.

Review Questions

• What does the Evaluation Theorem state about calculating definite integrals?
• If $F(x)$ is an antiderivative of $f(x)$, how do you compute $\int_2^5 f(x) \, dx$?
• Why is it necessary to know both limits of integration when using the Evaluation Theorem?

Related terms

Definite Integral: The definite integral of a function over an interval [a,b] gives the net area under its curve between those points.

Antiderivative: An antiderivative of a function \( f \) is another function \( F \) such that \( F' = f \).

Fundamental Theorem of Calculus:

A theorem that links differentiation and integration, stating that differentiation and integration are inverse processes.