Double Integral Choosing Change of Variable

In summary, the problem involves evaluating the integral \int_0^{1} \int_y^{y+2} \sqrt{(x-y)}dxdy with a change of variable. The suggested method is to sketch the region and identify the boundary lines, which can then be used as the variables u and v when applying the jacobian.
  • #1
FallingMan
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Homework Statement


Evaluate the following integral with a change of variable of your choice.

[tex]\int_0^{1} \int_y^{y+2} \sqrt{(x-y)}dxdy[/tex]


The Attempt at a Solution



I'm supposed to choose a u and v that will simplify the integral, but I have no idea how to even start this.

I tried substituting u = x-y, but that doesn't even look like it would make my life simpler. If I did that, I'm not sure what I would choose for v.

Any tips to push me in the right direction would be appreciated.
 
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  • #2
Have you tried to sketch the region that the integral is bounded over. If you do that you should be able to write four equations for the boundary lines, there should be two pairs that look the same (just shifted) and those two pairs will be your u and v. After that just use the jacobian and it becomes a simple integration problem.
 

Related to Double Integral Choosing Change of Variable

1. What is a double integral?

A double integral is a type of integral in mathematics that involves finding the volume under a curved surface in a three-dimensional space. It is represented by the symbol ∫∫ f(x,y) dA, where f(x,y) is the function being integrated and dA represents the infinitesimal area element.

2. What is the purpose of changing variables in a double integral?

Changing variables in a double integral allows for a more efficient and accurate way of calculating the integral. By substituting a new set of variables, the integrand can be simplified and the limits of integration can be changed to better fit the new variables, making the integration process easier.

3. How do I choose the appropriate change of variables for a double integral?

The appropriate change of variables for a double integral depends on the shape and orientation of the region being integrated. Some commonly used substitutions include polar, cylindrical, and spherical coordinates. The choice of substitution should be based on simplifying the integrand and making the limits of integration more manageable.

4. What is the Jacobian determinant and how is it used in changing variables?

The Jacobian determinant is a measure of how a change of variables affects the volume element. In the context of double integrals, the Jacobian determinant is used to convert the infinitesimal area element dA in the original coordinates to the corresponding infinitesimal area element in the new coordinates. It is represented by the symbol |J|.

5. What are the benefits of using a change of variables in a double integral?

Using a change of variables in a double integral can make the integration process simpler and more efficient. It can also help to visualize the region being integrated and can lead to a better understanding of the underlying geometry and relationships between the variables. Additionally, certain integrals may only be solvable using a specific change of variables, making it a necessary tool in solving certain problems.

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