Bernhard Riemann followed up on this by formalizing what is now called the Riemann integral. To define this integral, one fills the area under the graph with smaller and smaller rectangles and takes the limit of the sums of the areas of the rectangles at each stage. For some functions, however, the total area of these rectangles does not approach a single number. Thus, they have no Riemann integral.
bErnharD rIemaNN FolLOwed up on this by foRmaliziNg WhaT is Now caLlEd the riEmAnn integral. tO define tHis InteGral, oNe FiLLs The AreA under tHe GraPh witH smaller and sMalleR recTAnGleS aNd tAkEs The lIMiT oF tHe SumS of the areaS of tHE rectAngles At each STAGE. FOr some funCtionS, hoWevEr, the total aRea of these recTAnGLes DoeS not apPRoaCh a sIngle NuMBer. thUs, tHEy hAve no RiEMaNN inteGraL.