Connecting Differentiation and Integration
The Fundamental Theorem of Calculus (FTC) establishes a crucial link between the two main branches of calculus: differentiation (finding rates of change) and integration (finding accumulated change or areas under curves). It demonstrates that these operations are inverse processes of each other.
Part 1: The Accumulation Function
The first part of the FTC states that if a function is continuous, then an 'accumulation function' defined as the integral from a constant to a variable of that function, is differentiable, and its derivative is the original function itself. This means integrating a rate and then differentiating returns you to the original rate function.
Part 2: Evaluating Definite Integrals
The second and more commonly applied part of the FTC provides a method to evaluate definite integrals. It states that the definite integral of a function over an interval can be calculated by finding any antiderivative of the function and evaluating it at the upper and lower limits of integration, then subtracting the results.
Significance in Science and Engineering
The FTC is foundational for solving problems involving accumulation, net change, and total quantities. It's extensively used in physics (e.g., calculating work done by a variable force, displacement from velocity), engineering (e.g., total volume flow, material stress), and economics (e.g., total cost from marginal cost).