Signed area

The blue area above the x-axis may be specified as positive area, while the yellow area below the x-axis is the negative area.

The integral of a real function can be imagined as the signed area between the ${\displaystyle x}$-axis and the curve ${\displaystyle y=f(x)}$ over an interval [a, b]. The area above the ${\displaystyle x}$-axis may be specified as positive (${\displaystyle +}$), and the area below the ${\displaystyle x}$-axis may be specified as negative (${\displaystyle -}$).[5] The negative area arises in the study of natural logarithm as signed area above the curve ${\displaystyle y=1/x}$ for ${\displaystyle 0<x<1}$, that is:[6]

$${\displaystyle \ln x=\int _{1}^{x}{\frac {dt}{t}}=-\int _{x}^{1}{\frac {dt}{t}}<0.}$$

In differential geometry, the sign of the area of a region of a surface is associated with the orientation of the surface.[7] The area of a set A in differential geometry is obtained as an integration of a density:

$${\displaystyle \mu (A)=\int _{A}dx\wedge dy,}$$

where dx and dy are differential 1-forms that make the density. Since the wedge product has the anticommutative property, ${\displaystyle dy\wedge dx=-dx\wedge dy}$. 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 the area under the hyperbola xy = 1, the density dx ∧ dy is positive for x > 1, but since the integral

$${\displaystyle \int _{1}^{x}{\frac {dt}{t}}}$$

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 a negative area, and the natural logarithm consequently is negative in this domain.

Mikhail Postnikov's 1979 textbook Lectures in Geometry appeals to certain geometric transformations – described as functions of coordinate pairs ${\displaystyle (x,y)}$ – to express "freely floating area elements".[8] A shear mapping is either of:

$${\displaystyle {\begin{aligned}(x,y)&\to (x,\ y+kx),\\(x,y)&\to (x+ky,\ y)\end{aligned}}}$$

for any real number ${\displaystyle k}$, while a squeeze mapping is

$${\displaystyle (x,y)\to (\lambda x,\ y/\lambda )}$$

for any positive real number ${\displaystyle \lambda }$. An area element is related ${\displaystyle (\thicksim )}$ to another if one of the transformations results in the second when applied to the first. As an equivalence relation, the area elements are segmented into equivalence classes of related elements, which are Postnikov bivectors.

Proposition: If ${\displaystyle (a_{1},b_{1})=(ka+\ell b,\ k_{1}a+\ \ell _{1}b)}$ and

${\displaystyle \delta =k\ell _{1}-\ell k_{1}\neq 0,}$ then ${\displaystyle (a,b)\thicksim (a_{1},\ ({\frac {1}{\delta }})b_{1}).}$

proof: ${\displaystyle (a,\ b)\thicksim (a+{\frac {\ell }{k}}b,\ b)}$ shear mapping

${\displaystyle \thicksim (ka+\ell b,\ {\tfrac {1}{k}}b)}$ squeeze mapping

${\displaystyle \thicksim (ka+\ell b,\ {\tfrac {1}{k}}b+{\frac {k_{1}}{k\delta }}(ka+\ell b))}$ shear mapping

${\displaystyle =(ka+\ell b,\ {\frac {k_{1}}{k\delta }}(ka+({\frac {\delta }{k_{1}}}+\ell )b)}$

${\displaystyle =(ka+\ell b,\ {\frac {1}{\delta }}(k_{1}a+\ell _{1}b))\ \ {\text{since}}\ \ \ell +{\frac {\delta }{k_{1}}}={\frac {k}{k_{1}}}\ell _{1}}$

${\displaystyle =(a_{1},\ {\frac {1}{\delta }}b_{1}).}$

• Vector area
• Volume form
• Signed distance
• Signed measure
• Signed volume

• Kleitman, Daniel. "Chapter 15: Areas and Volumes of Parallel Sided Figures; Determinants". Kleitman's Homepage. MIT Math Department Homepage.