The Fundamental Theorem of Calculus is one of the most amazing and important theorems in analysis. It is a non-trivial result that links the concept of area and gradient, two seemingly unrelated concepts.
Fundamental Theorem of Calculus
The first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.
Let be a continuous real-valued function defined on a closed interval . Let be the function defined, for all in , by
Then is uniformly continuous on , differentiable on the open interval , and for all in .
Let and be real-valued functions defined on such that is continuous and for all ,
If is Riemann integrable on , then