Evaluating Double Integrals: Switching the Order of Integration

In summary: So the integral becomes\int^1_0 \int^y_0 cos(x/y) dxdyIn summary, to evaluate the given double integral, the order of integration must be changed to integrate with respect to y first, then x, with the limits of integration being 0 to 1 for both variables. The resulting iterated integral is \int^1_0 \int^y_0 cos(x/y) dxdy.
  • #1
Alw
8
0

Homework Statement



Evaluate the double integral by changing the order of integration in the iterated integral and evalutating the resulting iterated integral.

Homework Equations



[tex]\int[/tex][tex]^{1}_{0}[/tex] [tex]\int[/tex][tex]^{1}_{x}[/tex] cos(x/y)dydx

The Attempt at a Solution



I know how to solve a double integral after I've switched the order of integration, I'm having trouble with the acutal switching part :confused: The book we are using has one example in it regarding this, and it isn't very clear. If anyone would mind walking me through how to switch the order of integration, that'd be great :smile:

Thanks in advance,
-Andy

edit: The text for the integrals didnt come out well, to make it more clear, its the integral from 0 - to - 1 and the integral
from x - to - 1
 
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  • #2
Why must you change the order? The integral is easily solvable as it is.
 
  • #3
Gib Z said:
Why must you change the order? The integral is easily solvable as it is.

Is it? I can't see how.

I know how to solve a double integral after I've switched the order of integration, I'm having trouble with the acutal switching part The book we are using has one example in it regarding this, and it isn't very clear. If anyone would mind walking me through how to switch the order of integration, that'd be great

It always helps if you sketch the area over which you're integrating. Notice that the limits in x are from 0 to 1.

So, the area over which you're integrating is bounded in the x direction by the lines x=0 and x=1. Also, since the limits in y are from x to 1, the boundaries in the y direction are the lines y=x and y=1. Can you sketch the area now? From this, can you figure out how to switch the order of integration?
 
  • #4
siddharth said:
Is it? I can't see how.

I read it wrong :( I seemed to read cos (y/x) >.<" Damn
 
  • #5
Ok, thanks! so if I'm not mistaken then, the new equation is:

[tex]\int[/tex][tex]^{1}_{0}[/tex] [tex]\int[/tex] [tex]^{y}_{0}[/tex] cos(x/y)dxdy ?
 
  • #6
Yes. In your original integral x ranged from 0 to 1 and, for each x, y ranged from x to 1. That is the triangle with vertices (0,0), (1,1) and (0, 1). In the opposite order, to cover that triangle, y must range from 0 to 1 and, for each y, x must rage from 0 to y.
 

1. What is the purpose of switching the order of integration in double integrals?

The purpose of switching the order of integration in double integrals is to simplify the integration process and make it easier to evaluate. It can also help in solving more complex integrals that may not be possible to solve using the original order of integration.

2. When should I consider switching the order of integration in a double integral?

You should consider switching the order of integration when the limits of integration for one variable are dependent on the other variable, or when the integrand is easier to integrate with the switched order.

3. How do I switch the order of integration in a double integral?

To switch the order of integration, you need to first identify the limits of integration for both variables. Then, you need to rewrite the integral with the switched order of integration, making sure to change the limits of integration accordingly.

4. Are there any limitations to switching the order of integration in double integrals?

Yes, there are some limitations to switching the order of integration. It may not be possible to switch the order if the integrand is not continuous or if the original order of integration results in a simpler integral.

5. How can I determine if the order of integration has been successfully switched in a double integral?

You can determine if the order of integration has been successfully switched by evaluating the integral using both the original and switched orders. If the results are the same, then the order has been successfully switched. You can also use a graphing calculator or software to verify the results.

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