The Fundamental Theorem of Calculus

Understanding the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) serves as a pivotal link between the processes of differentiation and integration, two of the main operations in calculus. This theorem is divided into two parts, each highlighting a unique aspect of their interplay. The first part of the FTC establishes that if a function f is continuous on a closed interval [a, b], then the function g, defined by the integral of f from a to x (g(x) = ∫ from a to x of f(t) dt), is not only continuous on [a, b] but also differentiable on the open interval (a, b), with its derivative g'(x) being equal to f(x). This part confirms that differentiation can reverse the process of integration. The second part, often referred to as the "evaluation theorem," provides a method for calculating the definite integral of a continuous function. It states that if F is any antiderivative of a continuous function f on [a, b], then the integral of f from a to b is given by F(b) - F(a), allowing for the computation of the area under the curve of f between a and b.

The Derivative of the Integral Function

Exploring the first part of the FTC further, consider a continuous function f on a closed interval [a, b]. We define a new function F(x) as the integral of f from a to x. This function F is continuous on [a, b] and differentiable on (a, b). The essence of the theorem is demonstrated by showing that the derivative of F(x) is the original function f(x). This is established by examining the limit of the difference quotient of F as h approaches zero and employing the Mean Value Theorem for Integrals. The conclusion is that the process of integration, represented by the function F, can be reversed by taking its derivative, which yields the original function f(x). This result is significant as it solidifies the concept that integration and differentiation are inverse operations.

Evaluating Definite Integrals Using Antiderivatives

The second part of the FTC greatly facilitates the evaluation of the area under a curve for a continuous function f on a closed interval [a, b]. If an antiderivative F of f is known, the definite integral of f from a to b can be efficiently calculated by the difference F(b) - F(a). This theorem is invaluable in practice, as it bypasses the need for more intricate geometric or numerical approaches. The proof involves the function g(x) = ∫ from a to x of f(t) dt and the realization that any antiderivative F of f can be written as F(x) = g(x) + C, where C is a constant. Evaluating g at the endpoints a and b and using the properties of antiderivatives leads to the straightforward result that the integral of f from a to b is F(b) - F(a).

Practical Applications of the Fundamental Theorem of Calculus

The FTC is not only theoretically significant but also has extensive practical applications across various scientific and engineering disciplines. It is instrumental in the precise calculation of integrals, which are essential in determining areas, volumes, and other physical quantities that can be expressed as integrals. For example, the FTC is used to calculate the center of mass of an object, analogous to finding the balance point when standing on one leg. The center of mass is the point that represents the average location of the mass in a body, and the FTC aids in determining this point for objects with complex shapes and varying density distributions.

Examples Illustrating the Fundamental Theorem of Calculus

To demonstrate the FTC in action, consider several examples. Applying the first part of the FTC, we can differentiate a function defined by an integral with a variable upper limit. For instance, the derivative of g(x) = ∫ from 1 to x of (1/t^2 + 1) dt is simply 1/(x^2 + 1). In cases where the Chain Rule is required, such as for g(x) = ∫ from 1 to x^2 of cos(r) dr, the derivative is 2x·cos(x^2). When both limits of integration are functions of a variable, as in g(t) = ∫ from t to t^3 of s^4 ds, the integral is evaluated by applying the FTC to each part separately. Using the second part of the FTC, we can directly evaluate definite integrals, such as calculating the area under the curve of 4 - x^2 from 0 to 2, which results in 16/3. These examples showcase the FTC's utility in a wide array of integration problems.