The graph of a function in and of itself represents a pure number. It has no units. Consequently, the area under the graph is also a unitless number.
If, however, you are modelling something so that a unit is attached to the number represented by the function, , then the area under the graph also has a unit attached to it, namely the square of the unit attached to the function itself.
For example, if the function is to be interpreted as so many meters, then the area under the graph represents square meters.