On Non-Absolute Integrals

The Riemann integral, McShane integral and Henstock integral are well-known. They are the Riemann approach integrals. In this note we introduce the Riemann $\mathcal B$ -integral using Riemann approach, which are non-absolute integrals. The Fundamental Theorem of Calculus is proved. The ``local Character plays very important role in the prove of the Fundamental Theorem of Calculus. Some convergence theorems are also proved.