So I am sure I just have the syntax wrong but here is the situation....
I have Sheet2 with 3 columns: point_name, x, y I have Sheet3 with a list of areas with 5 columns:
Using your provided sample data, the results would be "Area1", "Area2", and "Area4". Those results are achieved using this formula:
=INDEX(Sheet3!$A$2:$A$5,MATCH(1,INDEX((Sheet3!$D$2:$D$5>=B2)*(Sheet3!$E$2:$E$5<=B2)*(Sheet3!$B$2:$B$5>=C2)*(Sheet3!$C$2:$C$5<=C2),),0))