if temperature is increased but the volume contained in a area, will an air balloon lift?
I case of an hot air balloon, the volume of the balloon remains unchanged when the air is heated. But because the heated air expands, part of it escapes from the balloon since it is not closed (the bottom is open). Therefore the weight of the air inside the balloon is reduced. The hot air in the balloon has a lower density that the cool air around it. Therefore the balloon lifts. 

 

Would a closed balloon of unstretchable fabric be used, then the pressure in the balloon would rise but the volume of the balloon would remain unchanged. Therefore the density of the air inside the balloon and the buoyancy would remain the same and the balloon would not lift off.