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