Future lightweight and long-focal-length x-ray telescopes must guarantee a good angular resolution (e.g., 5 arc sec HEW) and reach an unprecedented large effective area. This goal can be reached with the slumping of borosilicate glass sheets that allow the fabrication of lightweight and low-cost x-ray optical units (XOU). These XOUs, based on mirror segments, have to be assembled together to form complete multishell Wolter-I optics. The technology for the fabrication and the integration of these XOUs is under development in Europe, funded by European Space Agency, and led by the Brera Observatory (INAF-OAB). While the achievement of the required surface accuracy on the glass segments by means of a hot slumping technique is a challenging aspect, adequate attention must be given to the correct integration and coalignment of the mirror segments into the XOUs. To this aim, an innovative assembly concept has been investigated, based on glass reinforcing ribs. The ribs connect pairs of consecutive foils, stacked into a XOU, with both structural and functional roles, providing robust monolithic stacks of mirror plates. Moreover, this integration concept allows the correction of residual low-frequency errors still present on the mirror foil profile after slumping. We present the integration concept, the related error budget, and the results achieved so far with a semi-robotic integration machine especially designed and realized to assemble slumped glass foils into XOUs.