Overview
The Fundamental Theorem of Calculus connects the concept of the derivative of a function with the concept of the integral. It serves as a central unifying theorem in calculus and has two main parts.
The First Part
This part states that if a function is continuous on the closed interval [a, b], and F is the indefinite integral of f on [a, b], then F is differentiable on the open interval (a, b), and F' = f.
The Second Part
This part allows us to compute the integral of a function using its antiderivative. More formally, if F is an antiderivative of f on an interval I, then the following holds for all a, b ∈ I:
\[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]
Applications
The Fundamental Theorem of Calculus is essential in understanding the accumulation of quantities, such as area under a curve, displacement, and total change.