# Exchanging order of integration

• Nikratio
In summary, the conversation involves a person questioning the ability to exchange the order of integration for multiple integrals. They realize this cannot be done in certain cases and wonder what conditions allow for the exchange. They also mention that this concept is taught in Calculus I and depends on the shape of the area being integrated over. However, the specific integrals being discussed do not have limits or constants of integration, making them incorrect.
Nikratio
Hi,

For some reason I always believed that I can generally exchange the order of integration for multiple integrals, i.e.
$$\int \int f(x,y) \; dx \; dy = \int \int f(x,y) \; dy \; dx$$

However, I just had to realize that this cannot be true, since (with a a constant):
$$\int \int \frac{d a}{dx} \; dx \; dy = \int a \; dy = ay$$
$$\int \int \frac{d a}{dx} \; dy \; dx = \int \int 0 \; dy \; dx = 0$$

So I'm wondering under what conditions I can actually exchange the order of integration. I looked into a couple of Calculus books and they mostly mention in passing that it depends on the shape of the area that I'm integrating over. However, the integrals I'm concerned with are usually not over a real geometric area that I can visualize and reason about...

Nikratio said:
Hi,

For some reason I always believed that I can generally exchange the order of integration for multiple integrals, i.e.
$$\int \int f(x,y) \; dx \; dy = \int \int f(x,y) \; dy \; dx$$

However, I just had to realize that this cannot be true, since (with a a constant):
$$\int \int \frac{d a}{dx} \; dx \; dy = \int a \; dy = ay$$[/quote
$$\int \int \frac{d a}{dx} \; dy \; dx = \int \int 0 \; dy \; dx = 0$$

So I'm wondering under what conditions I can actually exchange the order of integration. I looked into a couple of Calculus books and they mostly mention in passing that it depends on the shape of the area that I'm integrating over. However, the integrals I'm concerned with are usually not over a real geometric area that I can visualize and reason about...
You need to go back and review Calculus I. You have put no limits of integration on these nor have you the constant of integration. Neither of the integrals you have above is correct.

## What is exchanging order of integration?

Exchanging order of integration is a method used in calculus to evaluate double or triple integrals. It involves switching the order of integration, either from integrating with respect to one variable first to integrating with respect to the other variable first, or vice versa.

## Why is exchanging order of integration useful?

Exchanging order of integration can make it easier to evaluate integrals, especially when the integrand is complicated or when the region of integration is difficult to define. It can also help to simplify the calculation process.

## What are the steps to exchange the order of integration?

The general steps for exchanging the order of integration are: 1) Draw the region of integration and identify the limits of integration, 2) Set up the integral in the original order, 3) Rewrite the integrand in terms of the other variable, 4) Change the limits of integration to correspond with the new order, and 5) Evaluate the integral in the new order.

## Are there any conditions for exchanging the order of integration?

Yes, there are certain conditions that must be met in order to exchange the order of integration. The integral must be absolutely convergent, and the region of integration must be defined by a finite number of boundaries. Additionally, the integrand must be continuous over the region of integration.

## Can exchanging order of integration change the value of the integral?

No, exchanging the order of integration does not change the value of the integral as long as the conditions mentioned before are met. It simply rearranges the order in which the integration is performed, but the final result should be the same.

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