Final answer:
The Fundamental Theorem of Calculus connects differentiation and integration, showing they are inverse processes. It has two parts: the first states that the derivative of an integral function equals the original function; the second says the integral of a function over an interval can be calculated using its antiderivatives evaluated at the interval's endpoints.
Step-by-step explanation:
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects the concept of differentiation with that of integration, establishing that they are essentially inverse processes. It consists of two parts: The First Part guarantees that if we have a continuous function f(x) on an interval [a, b], and define a function F(x) to be the integral of f from a to x, then F(x) is differentiable on (a, b), and its derivative is the original function f(x). Symbolically, this can be expressed as:
F'(x) = f(x) for all x in (a, b).
The Second Part states that if we have a continuous function f(x) on [a, b] and F(x) is any antiderivative of f (that is, F'(x) = f(x)), then the definite integral of f from a to b can be obtained by evaluating the antiderivative F at the endpoints and taking the difference:
∫ f(x) dx = F(b) - F(a)
The theorem is powerful because it provides a practical way to evaluate definite integrals without performing the limit processes that define them. Instead, we find an antiderivative and use it to calculate the integral. This principle is fundamental to both differential calculus and integral calculus, describing the relationship between derivatives and integrals, and is widely applied across various fields like physics, engineering, and economics.