What is the First Fundamental Theorem of Calculals (FTC1) and how does it relate differentiation and integration?

The First Fundamental Theorem of Calculus (FTC1) is a crucial connection between differentiation (differential calculus) and integration (integral calculus). It consists of two parts, each providing a reversible relationship between these two concepts.

Part 1: Given a function f is continuous on the closed interval [a, b], and F(x) is a function defined as the integral of f from a to x, then F'(x) (the derivative of F with respect to x) is equal to f(x).

In other words, if you have a function f(x) and you find its antiderivative F(x) by integrating f(x) with respect to x, then the derivative of F(x) is the original function f(x). This part shows that differentiation and integration are inverse operations.

Part 2: Given f is continuous on the interval [a, b], and F(x) is any antiderivative of f(x) on (a, b), then the definite integral of f(x) from a to b is equal to F(b) - F(a).

This part tells us that the definite integral of a function f(x) from a to b is the change in the value of any antiderivative F(x) of f(x) from x = b to x = a. This is often referred to as the "Fundamental Theorem of Calculus," although it's technically the first part of the theorem that establishes this result.