Changing the order of integration

In summary, changing the order of integration is a useful technique for simplifying the integration process when the original order is complex. The process involves rewriting the integral in terms of the other variable and considering the new limits of integration. It can affect the outcome of the integral, but there are restrictions and limitations, such as keeping the region of integration the same and ensuring the integrand is continuous and integrable. It is most commonly used for double and triple integrals, and may not be applicable to all types of integrals.
  • #1
Amaelle
310
54
Homework Statement
Calculate the following integral (look at the image)
Relevant Equations
Double integrals
Greetings!
As mentionned my aim is to change the order of integral, and I totally agree with the solution I just have one question:
as you can see they have put
0<=y<=1 and 0<=x<=y^2
but would it be wrong if I put
0<=y<=1 and y^2<=x<=1?
Thank you!
1622822248789.png
 
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  • #2
Yes, it would be wrong. That would be the other part of the square in your figure.
 
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Likes Delta2 and Amaelle
  • #3
Orodruin said:
Yes, it would be wrong. That would be the other part of the square in your figure.
I got it now, thanks a lot!
 

Related to Changing the order of integration

1. What is the purpose of changing the order of integration?

Changing the order of integration allows us to evaluate a double or triple integral in a different way, which can sometimes make the integral easier to solve. It also allows us to switch between integrating with respect to different variables, which can be useful in certain situations.

2. When should I consider changing the order of integration?

You should consider changing the order of integration when the original integral is difficult to evaluate or when the region of integration is more easily described in terms of different variables. It can also be helpful when trying to set up a double or triple integral to represent a specific volume, area, or mass.

3. How do I change the order of integration?

To change the order of integration, you can use the concept of Fubini's Theorem, which states that if the function being integrated is continuous, then the order of integration can be changed without changing the value of the integral. You can also use geometric reasoning and visualization to determine the new limits of integration.

4. What are some common mistakes to avoid when changing the order of integration?

One common mistake is forgetting to change the limits of integration when switching the order. It is important to carefully consider the new limits and make any necessary adjustments. Another mistake is not properly setting up the new integral, which can lead to incorrect solutions.

5. Are there any limitations to changing the order of integration?

Yes, there are some limitations to changing the order of integration. For example, if the function being integrated is not continuous, then Fubini's Theorem cannot be applied and the order cannot be changed. Additionally, the region of integration must be well-defined and the new limits must still cover the entire region. It is important to carefully consider these limitations before attempting to change the order of integration.

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