# If one region is above the $x$-axis, and the other below it, should “the ratio of their areas” use absolute value?

Good morning mathematicians,

I apologize for asking such a basic question. When you are to find the ratio of "Area 1" and "Area 2", such that "Area 1" is the area of a region above the x-axis and "Area 2" is the area of a region below the x-axis, are you looking for an absolute value of the ratio, or do you leave the value negative?

• It depends on context. After all, treating below-the-$x$-axis regions as having negative area is a key element in how integration works, so "maybe" the ratio here should be considered negative if this is an integration exercise. On the other hand, if you simply happen to be using integration as a means of computing the areas of a couple of geometric figures, one of which just happens to be below the $x$-axis, then it's likely that "area" should be taken in absolute value. Authors sometimes avoid confusion by writing, say, "the ratio of the signed areas". – Blue Jan 13 at 8:28

## 1 Answer

Typically, areas are always non-negative numbers, so a ratio of areas should also be non-negative (i.e. if the result is negative, the absolute value is taken).