Fundamental Theorem of Calculus

Definition

The Fundamental Theorem of Calculus links the concept of differentiation and integration. It states that if a function is continuous over an interval, then its integral can be computed using its antiderivative.

5 Must Know Facts For Your Next Test

1. The Fundamental Theorem of Calculus has two parts: the first part relates the derivative to the integral, and the second part provides a way to evaluate definite integrals.
2. The first part states that if $F$ is an antiderivative of $f$ on an interval $[a, b]$, then for every $x$ in that interval, $\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$.
3. The second part states that if $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$.
4. This theorem confirms that differentiation and integration are inverse processes.
5. Understanding both parts is crucial for solving problems involving definite integrals and their applications.

Review Questions

• What are the two main parts of the Fundamental Theorem of Calculus?
• How does the first part of the theorem relate differentiation to integration?
• In what way does the second part help in evaluating definite integrals?

Related terms

Antiderivative: A function whose derivative is the given function.

Definite Integral: The evaluation of an integral within specific limits, providing a numerical value.

Continuity: A property of a function if it does not have any breaks, jumps, or points where it is undefined within a given interval.