The Fundamental Theorem of Calculus connects differentiation and integration in two main parts:

1. First Part: It states that if ( f ) is a continuous real-valued function defined on a closed interval ([a, b]), and ( F ) is an antiderivative of ( f ) on ([a, b]), then: [ \int_a^b f(x) , dx = F(b) - F(a) ] This part establishes a way to evaluate the definite integral of a function when an antiderivative is known.

2. Second Part: It states that if ( f ) is a continuous real-valued function on an interval ([a, b]), then the function ( F ) defined by: [ F(x) = \int_a^x f(t) , dt ] is continuous on ([a, b]), differentiable on the open interval ((a, b)), and its derivative ( F'(x) ) is ( f(x) ). In other words, differentiation undoes the process of integration.

Together, these parts show that integration and differentiation are inverse processes - integration can be used to find the area under the curve of a derivative (Part 1), and differentiation can recover the original function from its integral (Part 2).