What is the fundamental theorem of calculus and its role in evaluating definite integrals?

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is a crucial concept that connects two key operations in calculus: differentiation (finding a derivative) and integration. It consists of two parts, each providing a relationship between these two operations.

Part 1: First Fundamental Theorem of Calculus (FTC1)

Given a function $f(x)$ that is continuous on the interval $[a, b]$, and a function $F(x)$ defined as the integral of $f(x)$ from $a$ to $x$, i.e., $F(x) = \int_a^x f(t) \, dt$, then $F'(x)$, the derivative of $F(x)$, is equal to $f(x)$. In other words, differentiation and integration are inverse operations.

Formally, FTC1 states that if $F(x) = \int_a^x f(t) \, dt$, then $F'(x) = f(x)$.

Part 2: Second Fundamental Theorem of Calculus (FTC2)

FTC2 provides a method for evaluating definite integrals. Given a function $f(x)$ that is continuous on the interval $[a, b]$, and a function $F(x)$ that is continuous on $[a, b]$ and differentiable on $(a, b)$, with $F'(x) = f(x)$, then the definite integral of $f(x)$ from $a$ to $b$ is equal to $F(b) - F(a)$.

Formally, FTC2 states that if $f$ is continuous on $[a, b]$ and $F$ is continuous on $[a, b]$ and differentiable on $(a, b)$ with $F'(x) = f(x)$, then $\int_a^b f(x) \, dx = F(b) - F(a)$.