Evaluation theorem

Definition

The Evaluation Theorem is a key part of the Fundamental Theorem of Calculus. It states that the definite integral of a function over an interval $[a, b]$ can be found using its antiderivative.

5 Must Know Facts For Your Next Test

1. The Evaluation Theorem connects differentiation and integration.
2. If $F(x)$ is an antiderivative of $f(x)$, then $\int_{a}^{b} f(x)\, dx = F(b) - F(a)$.
3. The theorem simplifies the process of finding areas under curves.
4. It is essential for solving problems involving definite integrals.
5. Knowing how to find antiderivatives is crucial for applying this theorem.

Review Questions

• What does the Evaluation Theorem state about the relationship between definite integrals and antiderivatives?
• How do you apply the Evaluation Theorem to compute $\int_{1}^{3} x^2\, dx$?
• Why is it important to find the antiderivative when using the Evaluation Theorem?

Related terms

Definite Integral:

The integral of a function over a specific interval $[a, b]$, which represents the net area under the curve between these points.

Antiderivative:

A function $F(x)$ whose derivative is equal to the original function $f(x)$. In other words, if $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.

Fundamental Theorem of Calculus:

A principle that links differentiation and integration, comprising two parts: one that defines the integral as an antiderivative and another that relates definite integrals to evaluations at boundary points.