What Is an Example of an Ambiguous Integral Due to Cancellation of Areas?

[chjopl]

Give an example of an integral on (-infinity, infinity) that will lead to an ambigious answer if we evaluate the interal in terms of cancellation of areas.

 

[HallsofIvy]

Hmmm, sounds like you want a function that is NOT integrable from 0 to infinity and is an odd function. ( f(x)= x leaps to mind.) Do you see WHY that leads to an "ambiguous answer"?

 

[wais]

The cancellation of areas is a concept in mathematics that refers to the cancellation of positive and negative areas in an integral, resulting in an ambiguous answer. This can occur when the function being integrated has both positive and negative values over the interval of integration.

For example, let us consider the integral of the function f(x) = x^2 on the interval (-infinity, infinity). This integral can be evaluated using the fundamental theorem of calculus as follows:

∫_-∞^∞ x^2 dx = [(x^3)/3]_-∞^∞ = ∞ - (-∞) = ∞ + ∞

Here, we can see that the positive and negative areas cancel out, resulting in an ambiguous answer of infinity plus infinity. This is because the function f(x) = x^2 has both positive and negative values over the entire real line, leading to the cancellation of areas.

In situations like this, it is important to carefully consider the properties of the function being integrated and the limits of integration to avoid the cancellation of areas. In this case, we could avoid the ambiguity by breaking up the integral into two separate integrals over the intervals (-∞, 0) and (0, ∞), where the function has only positive or negative values respectively.

In conclusion, the cancellation of areas can lead to an ambiguous answer when evaluating integrals. It is crucial to be aware of this concept and carefully consider the properties of the function and limits of integration to avoid any potential ambiguities.