Why must the area under a curve require a non-negative function? For context, this is the first math class in Bachelor's electrical engineering.
Our teacher has just given us a definition for the area under a curve as the following:

If $f(x)$ is an integrable and non-negative function on a closed interval $[a, b]$, then the area under the curve $y = f(x)$ in that range is the integral of $f(x)$ evaluated from $a$ to $b$.

I asked him why the non-negative condition is given, and we're a little stuck trying to find an answer.
In a real-life context, I understand that areas are negative. But when applied to stuff like physics, we'll have to account for the sign as it often denotes ideas like direction. Moreover, imo pure mathematics should allow for negative areas.
Does anyone have any answers?

This is a purely pedagogical gambit. The definition is phrased like that precisely to stop people like you asking awkward questions! Later, presumably, your course will introduce the idea that the area under the x-axis must be counted as negative. But for now, they are keeping it as simple as possible for people who may be unfamiliar with mathematical terminology.
