Fundamental Theorem of Calculus, Part 2

Definition

The Fundamental Theorem of Calculus, Part 2 states that if $F$ is an antiderivative of $f$ on an interval $[a, b]$, then the definite integral of $f$ from $a$ to $b$ is equal to $F(b) - F(a)$. It links the concept of differentiation with that of integration.

5 Must Know Facts For Your Next Test

1. The theorem requires the integrand function to be continuous on the interval $[a, b]$.
2. It provides a way to evaluate definite integrals without having to compute Riemann sums.
3. If $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) \ dx = F(b) - F(a)$.
4. This theorem simplifies calculating areas under curves by using antiderivatives.
5. It connects indefinite integrals (antiderivatives) and definite integrals via evaluation at bounds.

Review Questions

Related terms

Antiderivative: A function whose derivative is the original function. For example, if $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.

Definite Integral: The evaluation of the area under a curve within given bounds. Represented as $\int_a^b f(x) \ dx$.

Indefinite Integral: Represents a family of functions and includes a constant of integration. Expressed as $\int f(x) \ dx = F(x) + C$, where $C$ is any constant.