pyqgis - Calculate area of all polygons in a shapefile and save it in attribute column "Area" in a Python script using QGIS

I have a shapefile with about 2000 polygons. I am using QGIS and I need to write a standalone Python script which opens the shapefile and calculates the area of all polygons separately and outputs the calculated area in a column "Area" in square kilometers. I tried to do it using the example in Calculate area in square meters from degrees?

eligibleLayer = QgsVectorLayer("/home/usr/Desktop/eligible_areas.shp", "eligible_areas", "ogr")

if not eligibleLayer.isValid():
    print "Eligible areas layer failed to load!" 

lProvider = eligibleLayer.dataProvider()
lProvider.addAttributes( [ QgsField("Area",QVariant.Double) ] )
area = 0 
for gFeat in eligibleLayer.getFeatures():
    calculator = QgsDistanceArea()
    calculator.setEllipsoid('WGS84')
    calculator.setEllipsoidalMode(True)
    geom = gFeat.geometry()
    landArea = gFeat.attributes()[gProvider.fieldNameIndex('Area')].toDouble()[0]
    if geom.isMultipart() is False: # if only simple polygon, calculate only for this

        polyg = geom.asPolygon() #transform to list of points
        if len(polyg) > 0:
            area = calculator.measurePolygon(polyg[0])

    else: #is Multipart
        multi = geom.asMultiPolygon()
        for polyg in multi:
            area = area + calculator.measurePolygon(polyg[0])

The script calculates the variable area but I am unsure if it is doing the right thing since I checked with the geometry tools and I get something completely different. Furthermore, I am unsure about how to update the above code such that I save the information in a newly created attribute column called "Area".