Upon the recommendation in writing by the district engineer, or upon a petition signed by a majority of the electors of the district owning land to be affected, or by electors representing at least one-half of the total acreage to be affected by such a proposed local improvement, defining the boundaries thereof, and requesting the board of directors to undertake the carrying out of the same, the board of directors, if it approves the same, may form and designate such area as an improvement district for the purpose desired, and shall establish and define the boundaries thereof.