I doubt that JoKo meant the area of a curve. The area bounded by various curves or lines was probably intended. Such an area can't be negative, regardless of whether or not the x-axis is part of the boundary.
For example: find the area of the region bounded by the x-axis, y= 0, and the parabola, \(\displaystyle y= x^2- 1\).
The first thing you should do is draw the graph- or at least imagine it- to see that the two cross at x= -1 and x= 1, that the only region bounded by those is between x= -1 and x= 1, and that, in that region, the x-axis is always above the parabola.
To find the area, subtract the equation of the lower curve from the equation of the higher curve, 0- (x^2- 1)= 1- x^2, and integrate from x= -1 to x= 1:
I have thanked so many posts here because I have learned something from this interesting question.
I must admit that I had always interpreted the "area under the curve" as meaning a number that was positive if the area was above the x-axis and negative if below the x-axis. That is, I was using the term in a figurative or perhaps conventional sense rather than a literal sense. Obviously an area must be positive, but a limit can be positive, negative, or zero.