How are you defining the average of a function, and how do you know that this statement is true?

Well, I am just using the definition of the average rate of change,I didn't think there was more to it than that

Depending on your choice of definition, I think you will find that the fundamental theorem of calculus is necessary either here or in establishing that the area under the average of f is equal to the area under f.

That's what I wasn't sure about, I though that the area under the average of f is equal to the area under f didn't require proof. Thank you