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?
Best Answer
Areas are not ever negative, no matter how pure the mathematics.
The statement in the question defines what the area under a curve is for the case of a non-negative and integrable function. It doesn't say anything about what the area under a curve might be defined to be if the function is negative or not integrable.
The point here is that the area under the curve is the area of the geometric figure defined by the $x$ axis and two parallel vertical lines $x=a$ and $x=b$ and the function value. If you want to define "area under the curve" for negative functions, you'll have to think of another definition for that, presumably the area between the function and the $x$ axis. But it won't ever be negative.
Regarding negative areas, let's draw an analogy with a map. Suppose you draw a north-south line through Hamburg, and then you say "areas west of this line are negative", so areas in Berlin are negative, and areas in Paris are positive. Then you can take the area of the Brandenburg gate and add it to the area of the Arc De Triomphe and get a region with zero area. That is clearly ridiculous, but it's no less ridiculous to claim that the region between the x axis and a negative function has a "negative area". The area of a bounded region is the amount of two-dimensional space it occupies, and that cannot ever become negative.