Change of variables in double integrals textbook problem

In summary, the conversation is about finding the area of a finite plane region bounded by four parabolas. The region is called D and is represented by the rectangle R in the uv-plane. The Jacobian determinant is used to calculate the area, but the individual expressions for u and v are not understood by the speaker.
  • #1
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There is an example in my textbook which I´m having trouble with.

The example is like this.
"
Find the area of the finite plane region bounded by the four parabolas, y=x^2 , y=2x^2 , x=y^2 , and x=3y^2
The region is called D.

Let u=y/x^2 and v=x/y^2

The the region D corresponds to the rectangle R in the uv - plane given by
1≤ u ≤2 and 1≤ v ≤3
"
The rest with the Jacobian determinant and so on I understand.

What I don't understand in this example is the expression
u=y/x^2 and v=x/y^2
I don't see why u and v are chosen this way.
It would be great if someone could explain this to me.
 
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  • #2
When u = y/x^2, y = u*x^2 and when v = x/y^2, x = v*y^2. So for example when y ranges from x^2 to 2x^2, u ranges from 1 to 2.
 
  • #3
I understand this better now.
Thanks
 

FAQ: Change of variables in double integrals textbook problem

What is a change of variables in double integrals?

A change of variables in double integrals is a method used to evaluate integrals over a region in the xy-plane by transforming the variables to a new set of coordinates, usually u and v. This allows for easier integration and can be used to solve more complex problems.

Why do we need to use a change of variables in double integrals?

A change of variables is used to simplify the integration process. It can be particularly useful when dealing with non-rectangular regions or when the integrand is difficult to integrate in the original coordinates. It allows for a more efficient evaluation of the integral.

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

The appropriate change of variables depends on the given problem and the region of integration. Some common techniques include polar coordinates, substitution, and Jacobian transformations. It is important to carefully analyze the problem and choose a change of variables that will make the integral easier to evaluate.

Can a change of variables be used for any type of double integral?

Yes, a change of variables can be used for any type of double integral. However, it may not always be necessary or beneficial to use one. It is important to consider the complexity of the integrand and the region of integration before deciding to use a change of variables.

Are there any limitations or restrictions when using a change of variables in double integrals?

Yes, there can be limitations or restrictions when using a change of variables. For example, some transformations may not be suitable for certain types of integrals or may result in a more complex integral. It is important to carefully evaluate the problem and choose an appropriate change of variables to avoid any potential limitations or restrictions.

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