# Change of variables in integration.

• vineel49
In summary, the original integral is transformed using the substitution x+y=u and x-y=v, resulting in a rotated coordinate system. The limits of the integrals are determined by considering the transformation and the Jacobian matrix, with one limit being from 0 to ∞ and the other being from a non-zero value to ∞.
vineel49

## Homework Statement

The original integral is
$$\left[\int_0^{\infty} {\int_0^{\infty} {F(x + y,x - y) \cdot dx \cdot dy} } \right]$$

What should be the limits of the integrals. (position represented by '?' symbol)
$$\left[\int_?^? {\int_?^? {F(u,v) \cdot (\frac{1}{2})du \cdot dv} } \right]$$ .

When x+y is substituted by 'u' and x-y is substituted by 'v'

## Homework Equations

use x+y= u , x-y=v, I am confused about the limits of the integrals

## The Attempt at a Solution

dx*dy = 0.5 * du * dv - This I got by using Jacobian matrix.
I need help in deciding the limits of the integrals.

My approach:
$$\left[\int_0^{\infty} {\int_{\left| v \right|}^{\infty} {F(u,v) \cdot (\frac{1}{2})du \cdot dv} } \right]$$

Is this correct?

Last edited by a moderator:
hi vineel49!
vineel49 said:
use x+y= u , x-y=v, I am confused about the limits of the integrals

try drawing it …

that's simply rotating the axes by 45°, isn't it?

(and scaling up or down by the Jacobian)

so if one of the new limits is 0 to ∞, the other must be … (not 0) to … ?

## 1. What is the purpose of using change of variables in integration?

Change of variables in integration is a technique used to simplify integrals by replacing the original variables with new ones. This can make the integral easier to solve or can help in evaluating integrals that are otherwise impossible to solve using traditional methods.

## 2. How do I know when to use change of variables in integration?

Change of variables is typically used when the integral involves a complicated integrand or when the limits of integration are not simple. It can also be used to transform an integral into a form that is easier to solve, such as converting a rational function into a trigonometric function. In general, if the integral seems difficult to solve using traditional methods, it may be worth considering change of variables.

## 3. What is the process for using change of variables in integration?

The first step is to identify the original variables and determine which new variables will be used to replace them. Then, a substitution is made in the integral to express it in terms of the new variables. This may involve using trigonometric identities, completing the square, or other techniques. After the substitution is made, the integral can be solved using traditional methods, and the result can be expressed in terms of the original variables.

## 4. Are there any special cases where change of variables is particularly useful?

Yes, there are several special cases where change of variables can be very helpful. For example, when dealing with integrals involving trigonometric functions, using the appropriate trigonometric substitution can often simplify the integral significantly. Additionally, in cases where the integrand contains a square root, using a suitable substitution can often eliminate the square root and make the integral easier to solve.

## 5. Can change of variables be used in multiple dimensions?

Yes, change of variables can be used in multiple dimensions, such as in double or triple integrals. In these cases, the substitution involves transforming the coordinates of the integral into a new set of coordinates. This can be useful in solving integrals involving complex shapes or regions, such as those in polar or spherical coordinates.

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