Double integrals + Change of variables

In summary: You could integrate it directly..Yes, because there are some problems which say evaluate the integral with change of variable but then check it by using an iterated integral. So the answer is yes, you can integrate it directly.
  • #1
babbagee
74
0
Ok, i have a problem with this double integral. I am having a hard time finding the limits. The question is

Evaluate
[tex]
\iint \frac{dx\,dy}{\sqrt{1+x+2y}}\
[/tex]

D = [0,1] x [0,1], by setting T(u,v) = (u, v/2) and evaluating the integral over D*, where T(D*)=D

Can some one help me find the limits, and explain the process of getting those limits.

Thanks in advance
 
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  • #2
Are you REQUIRED to do that change of variables?
You could integrate it directly..
 
  • #3
Yes, because there are some problems which say evaluate the integral with change of variable but then check it by using an iterated integral. So the answer is yes, i have go use change of variable, and even though i don't need to use it was to get practice at it.

Thanks
 
  • #4
anybody?
 
  • #5
Rajvirnijjar said:
anybody?
Somebody is always here but not your wishing somebody.
Can you find the integration (to x variable) of 1/root(a+x) ?
 
  • #6
I THINK what you are saying is that you want to use the substitution u= x, v= 2y.
Of course, du= dx and dv= 2dy or dy= (1/2)dv.

In terms of u and v, the integral becomes
[tex]\frac{1}{2}\int \frac{du\,dv}{\sqrt{1+u+v}}[/tex]

The only problem now is finding D*. The boundaries of D are x= 0, x= 1, y= 0, y= 1.
Okay, when x= 0 what is u? When x= 1, what is u? When y= 0, what is v? When y= 1, what is v? That gives you D* and the limits of integration.
 
  • #7
Oh well, sorry for my bad English, I didn't read the whole OP and thought he didn't know how to solve it, but he should say the same as you did anyway, I know that for certain :biggrin:
 
Last edited:
  • #8
Thanks

I thought about the problem a little harder and i did the same exact thing you did, so thanks for your help.
 
  • #9
What is the non-graphing method to find the new limits ? Someone please,,,,,...!
 

Related to Double integrals + Change of variables

1. What are double integrals and why are they important in mathematics?

Double integrals are a type of mathematical operation used to calculate the signed volume between a two-dimensional region and a function that may vary over that region. They are important in mathematics because they allow for the calculation of areas, volumes, and other quantities in multi-dimensional spaces, which has numerous applications in various fields such as physics, engineering, economics, and more.

2. How do you solve a double integral?

To solve a double integral, you first need to identify the bounds of integration, which are the limits of the region over which the integration will take place. Then, you need to set up the integral by multiplying the function by the differential area element (dx dy or dy dx) and integrating over the given bounds. Finally, you evaluate the integral using appropriate techniques such as u-substitution or integration by parts.

3. What is the change of variables method and when is it used in double integrals?

The change of variables method is a technique used to simplify the evaluation of double integrals by transforming the original variables into new ones. This is typically done to convert the boundaries of the region of integration to a more convenient form or to simplify the integrand. It is especially useful when dealing with non-rectangular regions or complicated functions.

4. How do you choose the appropriate change of variables for a double integral?

There are several techniques for choosing the appropriate change of variables for a double integral, including substitution, polar coordinates, and other coordinate transformations. The choice depends on the specific problem and the desired outcome, such as simplifying the integrand or transforming the region of integration into a more convenient form. It is important to carefully analyze the given problem and select the most suitable change of variables.

5. What are some common mistakes to avoid when working with double integrals and change of variables?

Some common mistakes to avoid when working with double integrals and change of variables include not carefully identifying the bounds of integration, not correctly setting up the integral, and not properly evaluating the integral. It is also important to be mindful of the order of integration and to carefully choose the appropriate change of variables for the given problem. It is always helpful to double-check your work and make sure you have correctly applied the necessary concepts and techniques.

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