Choosing the order of integration with double integration

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In summary, the conversation discusses changing the order of integration for a double integral and using the integration by parts method. The person asking for help is struggling with finding the new limits after changing the order and is looking for a way to eliminate the 2x term. The conversation also emphasizes the importance of sketching the region of integration before attempting to solve the integral.
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8614smith
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Homework Statement


Sketch the regions of integration, and evaluate the integral by choosing the best order of integration.
[tex]\int^{2\sqrt{ln2}}_{0}\int^{\sqrt{ln2}}_{y/2}exp(x^2)dxdy[/tex]


Homework Equations


integration by parts


The Attempt at a Solution


ive changed the order of integration and done the inner integral with respect to y to get this far..
[tex]\int^{\sqrt{ln2}}_{y/2}\int^{2\sqrt{ln2}}_{0}exp(x^2)dydx=\sqrt{ln2}\int^{2\sqrt{ln2}}_{y/2}exp(x^2)dx[/tex]

and now when i do a u substitution
[tex]u=x^2[/tex]
[tex]du=2xdx[/tex]
and that is how far i can get as i can't put the 'du' into the equation as it has a '2x' in it and i will be integrating with respect to 'u'. Integrating it the original way round just gets me to this problem straight away.

I just can't see any way of getting rid of the 2x?!
 
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  • #2
If you change the order by which you integrate, the limits will change. You need to find the new limits.

Also ∫ex2dx does not exist in terms of elementary functions.
 
  • #3
Make sure you sketch the region of integration, as step that many students skip, believing it to be unimportant.
 
  • #4
that seems so obvious now but you wouldn't believe how long I've been looking at it, the 'cancelling 2x' comes from when you change the limits. thanks guys!
 

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

The order of integration determines the sequence in which the variables are integrated in a double integral. It is important because it affects the ease of computation and the final result of the integral.

2. How do you determine the appropriate order of integration for a given double integral?

The appropriate order of integration can be determined by considering the shape and limits of the region of integration. It is generally easier to integrate with respect to the innermost variable first.

3. Can the order of integration be changed for a double integral?

Yes, the order of integration can be changed for a double integral as long as the region of integration remains the same. However, the final result may differ depending on the new order of integration.

4. What is the difference between changing the order of integration and performing a change of variables in a double integral?

Changing the order of integration only rearranges the sequence in which the variables are integrated, while a change of variables involves substituting the original variables with new ones. Both methods can be used to simplify the integration process, but they are not interchangeable.

5. Are there any general guidelines for choosing the order of integration in double integration?

There are no strict rules for choosing the order of integration, but some general guidelines include integrating with respect to the variable with the simplest limits first, and choosing an order that allows for the easiest integration of the function. In some cases, symmetry and the type of function being integrated may also be considered.

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