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## Summary

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.

The integral of a real function can be imagined as the signed area between the line $y=0$ and the curve $y=f(x)$ over an interval [a, b].

Negative area arises in the study of natural logarithm as signed area under the curve y = 1/x for x in the positive real numbers:

Definition: $\ln x=\int _{1}^{x}{\frac {dt}{t}},\quad x>0.$ "For 0 < x < 1, $\ln x=\int _{1}^{x}{\frac {dt}{t}}=-\int _{x}^{1}{\frac {dt}{t}}<0$ 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: $\mu (A)=\int _{A}dx\wedge dy$ where dx and dy are differential 1-forms that make the density. Since the wedge product has the anticommutative property, $dy\wedge dx=-dx\wedge dy.$ 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 $\int _{1}^{x}{\frac {dt}{t}}$ 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.