The Riemann Integral

The Fundamental Theorem of Calculus is a statement about the inverse relationship between differentiation and integration. It comes in two parts, depending on whether we are differentiating an integral or integrating a derivative. Under suitable hypotheses on the functions f and F, the Fundamental Theorem of Calculus states that $$ \begin{array}{l} \left( i \right)\,\int_a^b {F'\left( x \right)} dx = F\left( b \right) - F\left( a \right)\;and\\ \left( {ii} \right)\,if\,G\left( x \right) = \int_a^x {f\left( t \right)} dt,\,then\,G'\left( x \right) = f\left( x \right). \end{array} $$