When is an integral infinite help

In summary, the problem is asking if the function f(x,y) = \frac{xy}{(x^2+y^2)^2} is integrable with respect to the Lebesgue measure on the set [-1,1]\times[-1,1]. The integral is infinite, meaning f is not integrable. However, the next problem asks to show that the double integrals exist, which seems contradictory. The function f is measurable, as it is only discontinuous at (0,0), and is defined on the entire square.
  • #1
P3X-018
144
0

Homework Statement


The problem is this:

Let f : R^2 -> R be

[tex] f(x,y) = \frac{xy}{(x^2+y^2)^2}\qquad (x,y)\neq(0,0) [/tex]

With [itex] f(0,0) = 0 [/itex].

Question:
Is [itex] f [/itex] integrable with respect to the Lebesguemeasure [itex] m_2 [/itex] om the set [itex] [-1,1]\times[-1,1] [/itex]?

The Attempt at a Solution



Well [itex]f[/itex] is integrable if and only is

[tex] \int_{[-1,1]^2} \vert f\vert\,dm_2(x,y) < \infty[/tex]

The integral is infinite, and so f isn't integrable.
The thing that makes it confusing, is that in the next problem it says,
Show that both the double integrals exist

[tex] \int_{[-1,1]}\left(\int_{[-1,1]} f(x,y)\,dm(x)\right)dm(y)[/tex]
[tex] \int_{[-1,1]}\left(\int_{[-1,1]} f(x,y)\,dm(y)\right)dm(x)[/tex]

But how can they exist if f isn't integrable?


EDIT:
Hmm I had made an error. It did seem kinda wired that the integral of a positive would give me something like [itex] \infty-\infty[/itex]. The integral is infinite though. The expression I had found would only hold for y > 0.
 
Last edited:
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  • #2
First question : Dont you think that the integral should have two boundaries? its from [-1,1] to what? please try to be more specific so we can help
 
  • #3
Actually [-1,1] is not a point it's the interval from -1 to 1. That's just another way of writting the integral of the function f(x,y) from -1 to 1 with respect to x (hence I have writtin m(x), the Lebesgue measure with respect x).
 
  • #4
I assume that the notation "[-1,1]2" means [itex]-1\le x\le 1[/itex] and [itex]-1\le y\le 1[/itex]. A function is as long as the set of points on which it is NOT continuous is a measurable set. For what points is this function not continuous?
 
  • #5
HallsofIvy said:
I assume that the notation "[-1,1]2" means [itex]-1\le x\le 1[/itex] and [itex]-1\le y\le 1[/itex].

Yep, that's the square I'm talking about.

HallsofIvy said:
A function is [what?] as long as the set of points on which it is NOT continuous is a measurable set. For what points is this function not continuous?

What is the function if it's not defined on a measurable set? Integrable or measurable?
The function is measurable though, since as f is only discontinuous at the point (0,0), and we just define f(0,0)=0, and make f defined on the whole square.
 

1. What does it mean for an integral to be infinite?

When an integral is infinite, it means that the area under the curve of the function being integrated is infinite. This can happen when the function being integrated has an unbounded growth or oscillates infinitely.

2. How do you determine if an integral is infinite?

To determine if an integral is infinite, you need to evaluate the integral using proper techniques, such as the limit comparison test or the integral test. If the integral diverges, or goes to infinity, then it is considered to be infinite.

3. Why is it important to understand infinite integrals?

Understanding infinite integrals is important because they can represent physical quantities, such as the area under a velocity-time graph, or mathematical concepts, such as the sum of an infinite series. It also allows us to calculate other important values, such as the area between curves or the volume of a solid.

4. How can an infinite integral be helpful?

An infinite integral can be helpful in understanding the behavior of a function, as well as in solving problems in physics, engineering, and other fields that involve integration. It can also provide insights into the convergence or divergence of a series.

5. Are there any real-life applications of infinite integrals?

Yes, there are many real-life applications of infinite integrals. For example, in physics, infinite integrals can be used to calculate the work done by a variable force or the center of mass of an object. In economics, they can be used to model continuous growth and decay in financial markets. In engineering, they can be used to analyze the stability of structures under varying loads. Overall, infinite integrals have numerous applications in various fields of study.

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