Multiple integral change of variables

In summary, the region R is defined by x^2 + y^2 = 1 and x^2 + y^2 = 2y, with x>=0 and y>=0. The integral of x*e^y over the region D in the uv-plane, corresponding to R, is being computed using a change of coordinates. The new limits of integration have not yet been determined, and the integrand may need to be changed to a function of u and v. Completing the square for the second condition may help in determining the limits, and solving for y in terms of u and v may aid in changing the variables of the integrand.
  • #1
BrownianMan
134
0
Let R denote the region inside x^2 + y^2 = 1 but outside x^2 + y^2 = 2y with x=>0 and y=>0. Let u=x^2 + y^2 and v=x^2 + y^2 -2y. Compute the integral of x*e^y over the region D in the uv-plane which corresponds to R under the specified change of coordinates.

I'm having trouble with this one. My first attempt at figuring out the new limits of integration yielded 0<=u<=1 and 0<=v<=u, which seems wrong to me. I'm also not sure how to change the integrand to make it a function of u and v.
 
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  • #2
I'm still trying to work out the new limits of integration myself, but I think it might help to complete the square for the second condition...

As for changing the function's variables, can you solve for y in terms of u and v? (Hint: What is u-v?)
 

FAQ: Multiple integral change of variables

What is the concept of "Multiple integral change of variables"?

Multiple integral change of variables is a mathematical technique used to simplify the evaluation of integrals in multiple dimensions. It involves substituting new variables in place of the existing ones to transform the integral into a simpler form.

Why is "Multiple integral change of variables" useful in scientific research?

Multiple integral change of variables allows for the evaluation of complex integrals in a more efficient and intuitive way. This makes it an essential tool for solving various problems in physics, engineering, and other scientific fields.

What are the different methods used in "Multiple integral change of variables"?

There are several methods for changing variables in multiple integrals, including the Jacobian transformation, polar coordinates, cylindrical coordinates, and spherical coordinates. Each method is suitable for different types of integrals and can simplify the evaluation process.

How do you determine the appropriate change of variables for a given integral?

The choice of the change of variables depends on the geometric features of the region of integration. The most common approach is to use the Jacobian transformation, which involves setting up a system of equations and solving for the new variables.

Can "Multiple integral change of variables" be applied to integrals with more than two variables?

Yes, multiple integral change of variables can be extended to integrals with any number of variables. However, the complexity of the transformation increases with the number of variables, and it may not always lead to a simpler form of the integral. In such cases, other techniques such as numerical methods may be more useful.

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