Is Boundedness a Necessity for Double Integral Proofs?

In summary, the conversation discusses the relationship between a function being defined on a rectangle and being bounded on that rectangle, specifically in the context of double integrals. It is stated that if the double integral of the function on the rectangle exists, then the function must be bounded on that rectangle. However, there may be alternative definitions of integration where boundedness is not necessary.
  • #1
engin
8
0
Show that if f is defined on a rectangle R and double integral of f on R
exists, then f is necessarily bounded on R.
 
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  • #2
If the function is defined on the rectangle the I can't see how it could be unbounded there. Do you mean almost everywhere defined? In the second case, the statement is wrong. Try

[tex]\int_0^1\int_0^1 \frac{y}{\sqrt{x}}dxdy[/tex]
 
  • #3
engin said:
Show that if f is defined on a rectangle R and double integral of f on R
exists, then f is necessarily bounded on R.

This will be true if you define the integral in terms of ordinary Riemann or Darboux definition. There are more general definitions where boundedness is not required.
 

FAQ: Is Boundedness a Necessity for Double Integral Proofs?

1. What is a double integral proof?

A double integral proof is a mathematical method used to determine the area under a 2-dimensional surface. It involves calculating the integral of a function over a region in the xy-plane.

2. How is a double integral proof different from a single integral proof?

A single integral proof is used to calculate the area under a curve in one dimension, while a double integral proof involves calculating the area under a surface in two dimensions.

3. What is the purpose of using a double integral proof?

A double integral proof is commonly used in physics and engineering to calculate quantities such as volume, mass, and center of mass. It is also used in statistics and economics to calculate probabilities and expected values.

4. What are the steps involved in a double integral proof?

The steps involved in a double integral proof are: 1) determining the bounds of the region in the xy-plane, 2) setting up the integral with appropriate limits of integration, 3) evaluating the integral using techniques such as substitution or integration by parts, and 4) interpreting the result in the context of the problem.

5. Are there any common mistakes to avoid when using a double integral proof?

Some common mistakes to avoid when using a double integral proof include: 1) incorrectly setting up the integral by choosing the wrong limits of integration or not accounting for discontinuities in the function, 2) forgetting to include the appropriate units in the final answer, and 3) making calculation errors during the evaluation of the integral.

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