A portion of the cost and expense of any improvement made under the provisions of this chapter including the engineering expenses and including the making of the estimate of the cost and expense of the improvement, also the cost of supervision and inspection of the work and the ascertaining of the ownership of the property included in the assessment district and the cost of advertising and publishing of all notices shall be shown upon the said assessment roll and the amount of such cost and expense as so shown shall be paid into the current expense fund of the city by the city treasurer out of the first collections made upon the said local improvement district. In making up the assessment roll, the city engineer shall show the amount of such cost and expense.