In mathematics, the signed area or oriented area of a region of an affine plane is its area with orientation specified by the ("plus" or "minus") sign. More generally, the signed area of an arbitrary surface region is its surface area with specified orientation. When the boundary of the region is a simple curve, the signed area also indicates the orientation of the boundary.
"For 0 < x < 1, and so ln x is the negative of the area,,,"
In differential geometry, the sign of the area of a region of a surface is associated with the orientation of the surface: "In addition to the area ... one may consider also signed areas of portions of surfaces; in this case the area corresponding to one of the two possible orientations is defined by [A(H), a double integral] while the area corresponding to the other orientation is −A(H)"
Area of a set A in differential geometry is obtained as an integration of a density: where dx and dy are differential 1-forms that make the density. Since the wedge product has the anticommutative property, The density is associated with a planar orientation, something existing locally in a manifold but not necessarily globally.
In the case of the natural logarithm, obtained by integrating area under the hyperbola xy=1, the density dx ∧ dy is positive for x>1, but since the integral is anchored to 1, the orientation of the x-axis is reversed in the unit interval. For this integration the (− dx) orientation yields the opposite density to the one used for x>1. With this opposite density the area, under the hyperbola and above the unit interval, is taken as negative area, and the natural logarithm consequently is negative in this domain.