# Changing the variable in multiple integrals

• kidsmoker
In summary, the conversation discusses solving an integral using a transformation and the Jacobian. The method used involves finding new limits for the integrals and calculating the Jacobian factor. The solution is found using the transformed vertices, resulting in a rectangle and a final answer of 4π^4/3.
kidsmoker

## Homework Statement

Evaluate

$$\int\int(x-y)^2sin^2(x+y)dxdy$$

taken over a square with successive vertices (pi,0), (2pi,pi), (pi,2pi), (0,pi).

## Homework Equations

$$I = \int\int_{K} f(x,y)dxdy = \int\int_{K'} g(u,v)*J*dudv$$

where J is the Jacobian.

## The Attempt at a Solution

Okay so I've just been learning this for the first time, so I may be doing it completely wrong!

I used the transformations u=x-y, v=x+y which give the Jacobian as 2.

Now i wasn't sure how to get the new limits for the integrals. What I did was apply the transformation above to give new vertices:

(pi,0) -> (pi,pi)
(0,pi) -> (-pi,pi)
(pi,2pi) -> (-pi,3pi)
(2pi,pi) -> (pi,3pi)

This gives a simple rectangle, so then i just wrote

$$I = 2*\int^{3\pi}_{\pi}\int^{\pi}_{-\pi}u^2sin^2(v)dudv = \frac{4\pi^{4}}{3}.$$

I wish this was right, but I've a feeling it's not :-(

Any help greatly appreciated!

Last edited:
The rectangle looks ok. But haven't you got the jacobian factor upside down?

Ah yeah, should be 1/2. Other than that though does my method look correct?

Thanks.

kidsmoker said:
Ah yeah, should be 1/2. Other than that though does my method look correct?

Thanks.

Looks ok to me.

## What is a multiple integral and how is it different from a single integral?

A multiple integral is an extension of a single integral that involves integrating a function over a two-dimensional or three-dimensional region. It is different from a single integral in that it involves integrating over more than one variable.

## Why would you want to change the variable in a multiple integral?

Changing the variable in a multiple integral can make the integration process easier or more efficient. It can also help to simplify the expression or reveal certain properties of the integral.

## What are the steps for changing the variable in a double integral?

The steps for changing the variable in a double integral are as follows:

1. Identify the limits of integration for the new variables.
2. Substitute the new variables into the integrand.
3. Determine the Jacobian of the transformation.
4. Multiply the integrand by the Jacobian.
5. Integrate the new expression over the new limits of integration.

## Does the order of integration matter when changing the variable in a double integral?

Yes, the order of integration does matter when changing the variable in a double integral. It can affect the limits of integration and the expression of the integrand, which can ultimately impact the final result.

## Are there any common mistakes to watch out for when changing the variable in multiple integrals?

Some common mistakes to watch out for when changing the variable in multiple integrals include forgetting to include the Jacobian, using incorrect limits of integration, or not considering the order of integration. It is important to double check your work and make sure all the steps are followed correctly.

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