When any improvement in a city of the first class is completed according to contract, it shall be the duty of the city engineer to carefully inspect the improvement and if the improvement is found to be properly done, such engineer shall accept the improvement and report his or her acceptance to the board of public works or mayor, who shall report the same to the city council with recommendation that the same be approved or disapproved. The city council may confirm or reject such acceptance. When the ordinance levying the tax makes the same due as the improvement is completed in front of or along any block or piece of ground, the city engineer may accept the same in sections from time to time, if found to be done according to the contract, reporting his or her acceptance as in other cases.