Wolfram Demonstrations Project

Generally speaking, the line integral of a function over a given curve is defined by selecting points on the curve, evaluating the function's values at these points, and then taking the limit of the evaluated expression as the number of selected points approaches infinity in an appropriate way.

The definition of the slanted line integral is of a different nature. The integration is performed over a domain on the left-hand side of the curve, which is bounded by the given curve and two lines which are parallel to the axes. Let

be an integrable function and let

be its cumulative distribution function,

. Let

be a given continuous and tendable curve (i.e., its tendency is defined for all its interior points). Suppose that

is uniformly tended (i.e., its tendency indicator vector is constant for all its interior points). Then the slanted line integral of

over

is defined as follows:

where

is the positive domain of the curve (i.e., the domain bounded by

and two lines that are parallel to the axes, such that

is on the left of

is the tendency of the uniformly tended curve

, and

and

are the tendencies of the curve at the points

and

respectively. The letter

in the notation of the slanted line integral stands for "Slanted". The curve

should be a subcurve of another curve

, to assure that the tendencies at the points

and

are well defined. In case the curve

is not uniformly tended (the tendency indicator vector at the subcurve

is not equal to the tendency indicator vector at the subcurve

), then the slanted line integral of

over

is defined as the sum:

, where each of the integrals on the right is calculated as a slanted line integral over a uniformly tended curve.

In this Demonstration you can drag the points

, flip the orientation of the curve, or vary the curvature of the subcurves

and

to see how these changes affect each of the parameters in the definition of the slanted line integral.

Contributed by: Amir Finkelstein

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Snapshot 1: Since the option "skip the point

" was chosen and the curve

is uniformly tended, then

is evaluated via the double integral over the domain

on the left of the curve

. The colors of the edges show their tendencies for uniformly tended edges (green for positively tended, red for negatively tended, and black for zero tended). The only role of the blue edges

and

is to determine the pointwise tendencies at the points

and

, respectively; these edges' tendencies do not affect the slanted line integral over

. The vertices

, and

are hollow since the cumulative distribution function

is not evaluated there; the vertex

is dark green since the coefficient of

there is

, and the disks of the vertices

are light red to denote that the coefficient of

at those points is

Snapshot 2: The change with respect to the first snapshot is that the "skip the point

" option is unchecked, which lets us illustrate the additivity of the slanted line integral operator; the discrete Green's theorem can be used to prove this.

Snapshot 3: When compared to the first snapshot, this one illustrates another property of the slanted line integral, which holds for any uniformly tended curve whose tendency is never zero:

; again the discrete Green's theorem can be used to prove this.

Snapshot 4: Here

and

merge, so they cancel:

; hence the vertex is black, which implies that the coefficient of

at the point is zero. The curvature of the lines

only affects the domains

at the definition of the slanted line integral, as long as their tendency is unchanged.

Snapshot 5: The intersection of the domains

is colored in dark yellow; in both the third and the fourth snapshots, although the "skip the point

" option was selected, both integrals

are depicted because the curve

is not uniformly tended; hence in order to calculate the slanted line integral over

, the definition insists on going through the point where the curve ceases to be uniformly tended (the tendency indicator vector changes), that is, the point

Snapshot 6: Here we see in what sense this integration method is a natural extension to the one in the discrete Green's theorem: the slanted line integral over a perpendicular corner agrees with the parameter

from that theorem. In this case, the slanted line integral is

, and indeed the parameter

for such corners is -1.

The name "slanted" is due to the fact that this integration method is affected by the rotation angle of the coordinate system, since the domain

and the calculation of the cumulative distribution function both depend on that angle. Hence, it is assumed that the coordinate system is static and given in advance, which is a reasonable assumption in computer applications. In this Demonstration the axes are parallel to the edges of the figure. The definition of the slanted line integral might seem awkward at first glance, but this integration method has interesting properties that resemble those of the line integral and it enables a broader and a more rigorous discussion of the integration method used in the discrete Green's theorem: it extends the integration method found in the discrete Green's theorem to more general types of domains, those that are not necessarily formed by a finite union of rectangles. In turn, this semidiscrete integration method results in a theorem that extends the discrete Green's theorem to more general domains.

More details regarding the definition of tendency are also available in a previously uploaded Demonstration. A small remark regarding notation: the slanted line integral is supposed to be denoted as depicted in this link.

The theory of semidiscrete calculus is given in [1].

Reference

[1] A. Finkelstein. "The Theory behind the 'Summed Area Tables' Algorithm: A Simple Approach to Calculus." (May 25, 2010) www.arxiv.org/abs/1005.1418.