Precision measurement of the integrated luminosity of the data taken by BESIII at center-of-mass energies between 3.810 GeV and 4.600 GeVSupported by National Key Basic Research Program of China (2015CB856700), National Natural Science Foundation of China (NSFC) (11125525, 11235011, 11322544, 11335008, 11425524), Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program, Joint Large-Scale Scientific Facility Funds of the NSFC and CAS (11179007, U1232201, U1332201) CAS (KJCX2-YW-N29, KJCX2-YW-N45), 100 Talents Program of CAS, INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology, German Research Foundation DFG (Collaborative Research Center CRC-1044), Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey (DPT2006K-120470), Russian Foundation for Basic Research (14-07-91152), U.S. Department of Energy (DE-FG02-04ER41291, DE-FG02-05ER41374, DE-FG02-94ER40823, DESC0010118), U.S. National Science Foundation, University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt and WCU Program of National Research Foundation of Korea (R32-2008-000-10155-0)