Absolute Value in a double integral

In summary, the problem is to evaluate the double integral of \sqrt{|y-x^{2}|} over the region \Omega = [-1,1] x [0,2]. The solution involves splitting the integral into two parts, \sqrt{x^{2}-y} and \sqrt{y-x^{2}}, but the main difficulty is determining the limits of the integral. After some exploration and trial and error, the limits are found to be -1 \leq x \leq 1 and 0 \leq y \leq 2.
  • #1
adm_strat
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[SOLVED] Absolute Value in a double integral

Homework Statement



If [tex]\Omega[/tex] = [-1,1] x [0,2], evaluate the double integral [tex]\int\int_{\Omega} \sqrt{|y-x^{2}|} dA[/tex] given that it exists.

Homework Equations



None

The Attempt at a Solution



I know that in order to integrate with the absolute value I have to split the integral into two parts: When [tex]x^{2} > y ---> \sqrt{x^{2}-y} [/tex] and [tex]y > x^{2} ---> \sqrt{y-x^{2}} [/tex]

I just can't get of the limits of the integral. Anyone have any advice on where to start or how to look at it to discover the limits? TIA
 
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  • #2
n/m, found them out.
 

Related to Absolute Value in a double integral

1. What is absolute value in a double integral?

Absolute value in a double integral refers to the numerical value of a function without considering its sign. It is represented by two vertical bars surrounding the function.

2. How is absolute value used in a double integral?

Absolute value is used in a double integral to calculate the total area between the function and the x-axis. It is also commonly used to find the average value of a function over a given interval.

3. What is the difference between a regular integral and a double integral with absolute value?

In a regular integral, the absolute value is not used and the area between the function and the x-axis can be positive or negative. However, in a double integral with absolute value, the area is always positive as the negative values are converted to positive values.

4. Can absolute value be used with any type of function in a double integral?

Yes, absolute value can be used with any type of function in a double integral. It is particularly useful when dealing with functions that have negative values, as it allows for easier calculation of the total area.

5. Are there any limitations to using absolute value in a double integral?

One limitation of using absolute value in a double integral is that it only considers the magnitude of the function and does not take into account its direction. This can result in loss of information and may not accurately represent the behavior of the function.

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