The fundamental theorem of calculus is a theorem in mathematics that establishes the powerful connection between the concepts of differentiation and integration. It is considered one of the most fundamental theorems in calculus and plays a crucial role in many areas of mathematics and physics.
The theorem states that if a function is continuous on a closed interval [a, b], and if F is an antiderivative of f, then the definite integral of f from a to b is equal to the difference between the values of F at the endpoints a and b. In other words, the theorem relates the process of integration (finding the area under a curve) to the process of differentiation (finding the slope of a curve).
The fundamental theorem of calculus is divided into two parts. The first part, also known as the first fundamental theorem, states that if a function is continuous on a closed interval [a, b] and F is an antiderivative of f on that interval, then the definite integral of f from a to b is equal to F(b) - F(a). The second part, the second fundamental theorem, states that if f is a continuous function on an open interval (a, b) and F is an antiderivative of f, then the derivative of the definite integral of f from a to x is equal to f(x).
This theorem provides a fundamental tool for solving a wide range of problems that involve finding areas, volumes, velocities, and other quantities in calculus. It also forms the basis for various integral calculus techniques, such as substitution and integration by parts.
