Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus consists of two main parts that establish a profound connection between the concepts of differentiation and integration, which are two core operations in calculus.
The first part states that if f is a continuous real-valued function defined on a closed interval [a, b], and F is an antiderivative of f on that interval, then the integral of f from a to b can be computed as:
∫ab f(x) dx = F(b) – F(a)
This part allows us to evaluate definite integrals by using antiderivatives, simplifying the process of integration significantly.
The second part asserts that if f is a continuous function on an interval [a, b], then the function F defined by:
F(x) = ∫ax f(t) dt
is continuous on [a, b], differentiable on the open interval (a, b), and its derivative is the original function:
F'(x) = f(x)
This part demonstrates that differentiation and integration are fundamentally inverse processes, reinforcing the relationship between these two pivotal concepts in calculus.
In essence, the Fundamental Theorem of Calculus not only provides a way to compute definite integrals but also highlights the interconnectedness of mathematical analysis, making calculus a powerful tool in both theoretical and applied mathematics.