Improper Multiple Integrals (2)

In summary, the problem at hand is to determine whether the improper integral of sin(x^2 + y^2) / ln(x^2 + y^2) over the region bounded by x = 0, y = 0, and x = y converges or diverges. The conversation suggests attempting to solve this by converting the integral to polar coordinates, but it is difficult to show convergence or divergence using this method. The function r*sin(r^2)/ln(r^2) appears to be oscillatory and tends to infinity as r increases. Therefore, it is challenging to find a definition of integral that would allow for a conclusion about the convergence or divergence of this integral.
  • #1
kingwinner
1,270
0
I find this to be a very tough problem:

1) Determine whether the improper integral I
∞ ∞
∫ ∫ [sin (x2 + y2) / ln(x2 + y2)] dxdy
2 2
converges or diverges.


All I can think of and try is by changing it to polar coordinates:

I=A+B where

A=
pi/4---
----- ∫ [sin (r^2) / ln(r^2)] r dr(dtheta)
2 -- 2/sin(theta)

B=
pi/2 -------
---------- ∫ [sin (r^2) / ln(r^2)] r dr(dtheta)
pi/4 ---2/cos(theta)


But how can I show that each of them converges (or diverges)?

Can someone please help me out? Thank you!:smile:
 
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  • #2
What can you say about r*sin(r^2)/ln(r^2)? I would say it is oscillatory and tending in amplitude to infinity as r->infinity. You would need a very liberal version of the definition of 'integral' to be able to say that that exists. What's your definition of integral? Can't you find anything in that definition that would allow to say "This diverges."? I wouldn't even try to actually compute an iterated integral, the function form fits polar coordinates but the boundaries don't.
 

1) What is an improper multiple integral?

An improper multiple integral is an integral that has one or more of the following characteristics: the limits of integration are infinite, the function being integrated is unbounded, or the integrand has a discontinuity within the interval of integration.

2) How do you evaluate an improper multiple integral?

To evaluate an improper multiple integral, you first need to determine if it converges or diverges. This can be done by examining the behavior of the integrand near the points of discontinuity or infinity. If the integral converges, it can be evaluated using standard integration techniques. If it diverges, it is typically split into two or more integrals and evaluated separately.

3) What is the difference between a type 1 and type 2 improper multiple integral?

A type 1 improper multiple integral has infinite limits of integration, while a type 2 has a discontinuity within the interval of integration. This difference affects the techniques used to evaluate the integral, but the overall concept is the same.

4) Can improper multiple integrals be applied to real-life situations?

Yes, improper multiple integrals are frequently used in real-life situations, particularly in physics and engineering. Examples include calculating the center of mass of an object with varying density, finding the area under a curve with infinite bounds, or determining the volume of a solid with a curved boundary.

5) Are there any special rules for solving improper multiple integrals?

Yes, there are a few special rules to keep in mind when solving improper multiple integrals. These include using a substitution to transform the integrand into a more manageable form, using symmetry to simplify the integral, and employing techniques such as partial fractions or integration by parts. It is also important to be aware of the conditions for convergence or divergence of the integral.

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