Note: The PrintTask requires an ArcGIS Server 10.1 Export Web Map Task. So make sure you've got that first.
Now, there is also an option that's just referred to as "export": http://resources.arcgis.com/en/help/arcgis-rest-api/index.html#/Export_Map/02r3000000v7000000/
1) This is not the same thing as the Export Web Map task.
2) It does not print the basemap, only the layers on your server.
3) It does allow you to specify a bounding box.