The Fundamental Theorem of Calculus

Definition

The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration, two core concepts in calculus.
Essentially, it tells us how to evaluate definite integrals without going through the rigorous process of finding limits of Riemann sums.

The theorem is divided into two fundamental parts:
- The First Fundamental Theorem of Calculus: If a function is continuous over an interval [a, b] and F is an antiderivative of this function over the interval, then the definite integral of the function from a to b equals F(b) - F(a).
- The Second Fundamental Theorem of Calculus: If …f… is a function that is continuous over an interval I containing a, and x is in I, then the function F, defined by the integral from a to x of …f…, is continuous over I and differentiable over the interior of I, and F’ = …f….

The Fundamental Theorem of Calculus provides the link between integration and differentiation, suggesting they are in some way opposite processes (the integral being the “anti-derivative”).
It allows us to evaluate definite integrals without resorting to limits, making the calculation of areas simpler and more efficient.

As an example, consider the integration of a function f(x) = 2x over the interval [1, 4].
Using the first part of the Fundamental Theorem, we first find the antiderivative F(x) = x^2. Then, we compute F(4) - F(1) to get the value of the integral.

The Fundamental Theorem of Calculus has broad applications in science, engineering, and statistics.
It is fundamental for computations in physics, where laws are often expressed as differential equations.
In business and economics, it is applied in analyzing and predicting market trends based on historical data.

A common error is forgetting that the Fundamental Theorem relates to the anti-derivative: specifically, the definite integral is equal to the difference in values of the anti-derivative at the interval’s endpoints.
Another frequently overlooked point is the requirement for continuity of the function over the interval being considered. If the function is not continuous, the result may not be as predicted by the theorem.

The First Fundamental Theorem gives a method for calculating definite integrals without needing to use limiting processes.
The Second Fundamental Theorem provides a practical way to divide (‘differentiate’) the generating integral function, returning the original function.
Both sections of the theorem are essential, jointly stating that differentiation and integration are inverse processes.