Reversing order of integration

In summary, the problem is that the integral is trying to take place over a region that is too small.
  • #1
Fusi
4
0

Homework Statement



Reverse the order of integration:

[tex]\int_{0}^{1}\int_{y^2}^{2-y} f(x,y) \ dx\ dy[/tex] This was on a test I took today. I was having trouble with it and was wondering how I should have gone about it?
 
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  • #2
Fusi said:

Homework Statement



Reverse the order of integration:

[tex]\int_{0}^{1}\int_{y^2}^{2-y} f(x,y) \ dx\ dy[/tex]


This was on a test I took today. I was having trouble with it and was wondering how I should have gone about it?

Sketch a graph of the region over which integration is taking place. In the integral above, the inner integral runs from x = y2 to x = 2 - y. The outer integral runs from y = 0 to y = 1. When you reverse the order of integration for this problem you will end up with two iterated integrals.
 
  • #3
So x=2-y and x=[itex]y^{2}[/itex] so y=2-x y=[itex]\sqrt{x}[/itex]

intersect at: [itex]2-x=\sqrt{x}[/itex] [itex]x=1[/itex] [itex]y=1[/itex]

y at 0 is (0,2)

I got confused because at y=1,x=1 each side can be explained differently so my first attempt at a solutions was:

[tex]\int_{0}^{1}\int_{0}^{\sqrt{x}} f(x,y) \ dy\ dx + \int_{1}^{2}\int_{2-x}^{1} f(x,y) \ dy\ dx[/tex]

I used some random logic in my head to get that and we've never seen anything like that in class before so I crossed it out and just wrote:

[tex]\int_{0}^{2}\int_{2-x}^{\sqrt{x}} f(x,y) \ dy\ dx[/tex]
 
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  • #4
Fusi said:
So x=2-y and x=[itex]y^{2}[/itex] so y=2-x y=[itex]\sqrt{x}[/itex]

intersect at: [itex]2-x=\sqrt{x}[/itex] [itex]x=1[/itex] [itex]y=1[/itex]

y at 0 is (0,2)

I got confused because at y=1,x=1 each side can be explained differently so my first attempt at a solutions was:

[tex]\int_{0}^{1}\int_{0}^{\sqrt{x}} f(x,y) \ dy\ dx + \int_{1}^{2}\int_{2-x}^{1} f(x,y) \ dy\ dx[/tex]
This isn't correct, but it's closer than what you have below. Here's what you should get:
[tex]\int_{0}^{1}\int_{0}^{\sqrt{x}} f(x,y) \ dy\ dx + \int_{1}^{2}\int_{0}^{2 - x} f(x,y) \ dy\ dx[/tex]
Fusi said:
I used some random logic in my head to get that and we've never seen anything like that in class before so I crossed it out and just wrote:

[tex]\int_{0}^{2}\int_{2-x}^{\sqrt{x}} f(x,y) \ dy\ dx[/tex]
 
  • #5
Yea I made a mistake on my last post. I actually did write my second bound as [itex]{0}\leq{y}\leq{2-x}[/itex]

We've never seen a problem like that before so I got so nervous I just crossed the entire answer out. The answer I decided to write down in the last couple seconds was actually:

[tex]\int_{0}^{2}\int_{\sqrt{x}}^{2-x} f(x,y) \ dy\ dx[/tex]

I kind of just tried to push the two integrals from my past answer together so I wouldn't have two iterated integrals.
 
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  • #6
You understand why a single integral won't work, right?
 
  • #7
Yea, I have a bad habit of writing something that I know can't work when I get nervous.
 

FAQ: Reversing order of integration

What is "Reversing order of integration"?

"Reversing order of integration" is a mathematical technique used to change the order in which a multiple integral is evaluated. It involves swapping the variables of integration and reversing the direction of integration.

Why is "Reversing order of integration" important?

"Reversing order of integration" is important because it allows us to evaluate multiple integrals in a more efficient and sometimes simpler way. It can also help in solving difficult integrals that may not be possible to evaluate in the original order of integration.

What are the steps involved in "Reversing order of integration"?

The steps involved in "Reversing order of integration" are as follows:1. Identify the type of integral (double, triple, etc.)2. Determine the limits of integration for each variable.3. Swap the variables of integration.4. Reverse the direction of integration.5. Evaluate the new integral using the new limits of integration.

Is "Reversing order of integration" always possible?

No, "Reversing order of integration" may not always be possible. It depends on the integrand and the limits of integration. In some cases, it may result in a different integral that cannot be evaluated.

Can "Reversing order of integration" be used for any type of multiple integral?

Yes, "Reversing order of integration" can be used for any type of multiple integral, including double, triple, and higher order integrals. However, the process may vary slightly depending on the number of variables involved.

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