Find the area of a map (Xkm x Ykm) from the left & right longitude and latitude

This is not an OpenStreetMap specific question and should better have been asked at a general GIS or geography Q&A site. (Edit: I see it was cross posted. Not generally a recommended practice as it wastes helpers' time.)

You can use PostGIS for this. PostgreSQL with the PostGIS extension is a database system but often handy to do quick geometry or geography calculations like this:

# select st_area(
     st_setsrid(
        st_makebox2d(
            st_makepoint(103.55487,1.490873),st_makepoint(103.740497, 1.585770)
        ),
     4326)::geography);

    st_area      
------------------
  216753707.811525
(1 row)

The area is returned in square metres.

This isn't a simple question, as the answer won't be a square as the Earth isn't flat, so the bottom edge will be a different length to the top edge. Additionally because of the spherical bulge the surface will be greater than the 'flat' area. I think this might answer your question if you're willing to approximate the Earth to a sphere.

Edit: Of course this won't be in the format Xkm x Ykm but give the area in square km, if you specify a value for R, the radius of your spherical Earth, in km, for which 6,371km is an approximation.