Double integral with absolute value

In summary, you were trying to integrate y - x2, but you forgot the function |y - x^2| is inside a square root. So, you divided the two integrals and found that y > x2 and y < x2.
  • #1
norbellys
5
1

Homework Statement



I am trying to evaluate double integral
∫∫D (|y - x2|)½

D: -1<x<1, 0<y<2

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:
y>x^2−−−>√y−x2
y>x^2−−−>√y−x2

I just can't get of the limits of the integral

(it is this same questions https://www.physicsforums.com/threads/absolute-value-in-a-double-integral.202157/ )

Please I been trying to set this up for hours how
 
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  • #2
norbellys said:

Homework Statement



I am trying to evaluate double integral
∫∫D (|y - x2|)½

D: -1<x<1, 0<y<2

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:
y>x^2−−−>√y−x2
y>x^2−−−>√y−x2
You've written the same thing twice. If y > x2, then |y - x2| = y - x2.
If y < x2, then |y - x2| = -(y - x2) = x2 - y.
norbellys said:
I just can't get of the limits of the integral
Region D is just a rectangle.
norbellys said:
(it is this same questions https://www.physicsforums.com/threads/absolute-value-in-a-double-integral.202157/ )

Please I been trying to set this up for hours how
 
  • #3
norbellys said:

Homework Statement



I am trying to evaluate double integral
∫∫D (|y - x2|)½

D: -1<x<1, 0<y<2

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:
y>x^2−−−>√y−x2
y>x^2−−−>√y−x2

I just can't get of the limits of the integral

(it is this same questions https://www.physicsforums.com/threads/absolute-value-in-a-double-integral.202157/ )

Please I been trying to set this up for hours how
I forgot the function |y - x^2| is inside a square root. When I divide the two integrals for y > x2, |y - x2| = y - x2.
and y < x2, |y - x2| = -(y - x2) = x2 - y. I really don't know what would be the limits of integration would it be -1 to what then the other other what to 1?
 
  • #4
norbellys said:
I forgot the function |y - x^2| is inside a square root. When I divide the two integrals for y > x2, |y - x2| = y - x2.
and y < x2, |y - x2| = -(y - x2) = x2 - y. I really don't know what would be the limits of integration would it be -1 to what then the other other what to 1?
It takes a little practice to figure out how to select the limits of a double integral. If you choose to do the dy integral first, for each location x, the dy integral is performed and you need to figure out the limits (that often depend on x.). Then the dx integral is done adding up all the strips that were solved individually (as a function of x) when you did the dy integral.
 
  • #5
Thanks I think I got it!
 
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Related to Double integral with absolute value

1. What is a double integral with absolute value?

A double integral with absolute value is an integral that computes the total area under a surface, taking into account both positive and negative values. It is represented by the symbol ∫|f(x,y)|dA, where f(x,y) is the function being integrated and dA represents the infinitesimal area element.

2. How is a double integral with absolute value calculated?

To calculate a double integral with absolute value, you first need to determine the bounds of integration for both x and y. Then, you can use the standard formula for double integrals, which involves evaluating the inner integral first and then plugging the result into the outer integral. Finally, you can use the absolute value function to take into account both positive and negative values.

3. What is the significance of a double integral with absolute value in science?

A double integral with absolute value is commonly used in physics and engineering to calculate the total amount of work done by a force over a given region. It is also used in probability and statistics to calculate the probability of an event occurring within a certain range.

4. Are there any special cases or exceptions when dealing with double integrals with absolute value?

Yes, there are some special cases to consider when working with double integrals with absolute value. One common exception is when the function being integrated is not continuous, which can lead to undefined or infinite values. Additionally, the bounds of integration may need to be adjusted if the function has singularities or discontinuities within the region of integration.

5. How does a double integral with absolute value relate to other types of integrals?

A double integral with absolute value is a type of multiple integral, which also includes triple and n-fold integrals. It is also related to single integrals, which only involve one variable. Double integrals with absolute value are often used in conjunction with single integrals to calculate the volume of three-dimensional objects.

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