Suppose you had two continuous functions that can be integrated, f(x) and g(x), and you wanted to find the area between those functions between some arbitrary values of x, x1 and x2. Assume it's an "easy" (the region being integrated is always either above or below the x-axis) question. So then you have to find the definite integral of f(x) - g(x) between x1 and x2 with respect to x, right?
Does it matter at all if you swap the places of f(x) and g(x)?
My answer would go as follows: No, since it's area, by right, the answer must be positive in the end, taking the absolute value of the definite integral. The only difference it would make is that the definite integral is negative, but it's the same magnitude as when you do it the "right" way around. That doesn't matter, since you're already taking the absolute value in the first place because it's area. This is distinct from the case where the question asks directly for the definite integral of f(x) - g(x) between x1 and x2 rather than the area between two curves from x1 to x2.
But for some reason, my lecturer says that I cannot do this kind of function-swapping willy-nilly, since the absolute value function is only used when "needed". But... it's area. It is, by definition, needed. It's the same magnitude anyway, and area is entirely magnitude anyway. Since when could area, the product of two lengths, be negative or have direction in general? (I think linear algebra has some concept that corresponds roughly to area, but it's a signed value, but I don't remember what it is. That's besides the point, though.)
I don't get her reasoning at all. Why does she think this way? Is there an odd gotcha that happens later on in Calculus where you can't do this and expect the correct answer?