In math, can area under the curve be negative when...

...you use the trapezoid rule?
I hope not.

I had a final today and a multiple choice question worth 4 marks asked "use the trapezoid rule to approximate the area".
I ended with the answer -10, but there was -10 and +10.
I decided that even though I got a negative number, area is always positive. So I answered +10.

So if I'm wrong I get 0/4 because it's MC, as opposed to getting 3/4 if it was a written question.

That's exactly right, always use the absolute value when finding area (unless evaluating a symmetric function, in which case the area is usually 0).

YES: If they just gave you a function and interval, and asked to find area under the curve, you can get a negative value when curve is below x-axis.

NO: If they give you a diagram with a shaded region and ask for area of shaded region, then answer will be positive, even if region is below x-axis.