A question for the more math-oriented among us:

On my recent Calc test, this problem was asked:

True-False: (Integral symbol from 0 to 2) (x - x^3) dx represents the area under the curve y = x - x^3 from 0 to 2. Explain your answer.

My answer verbaitum:

It depends: since x - x^3 dips below the x axis, the integral does ** not ** represent the area bounded by x = 0, x = 2, y = 0, and y = x - x^3. However, if you consider the area under the x axis to have negative height (not only is there no area under the curve, there is negative area under the curve), then it does.

The teacher circled “It depends”, wrote “Not true”, and gave me half credit. Today, I argued that there was nothing intrinsically impossible about negative area. I pointed out that all formulas given for area would accept a negative height and return an answer without blowing up or being inconsistent. I pointed out that if you buried a 10 foot pole, although it would remain 10 feet long, the tip of the pole would not be 10 feet high. Finally, I pointed out since we were dealing with infinite sums and complex numbers (concepts which had no analogue in the real world), so conceptualizing negative area shouldn’t be impossible.

So, am I right? Would any dopers with formal training in math care to support/refute me?