Changing the order of a triple integration

In summary, the purpose of changing the order of a triple integration is to simplify the integration process and potentially save time and effort. This can be done by rearranging the integrals using properties of integration or taking advantage of symmetry in the integrand. The benefits of changing the order include making the integration more manageable and revealing patterns and relationships between variables. However, there may be limitations in dealing with complex functions or regions of integration, and it is important to carefully consider the limits of integration. While the order can be changed for most types of functions, there may be cases where it is not possible or advantageous.
  • #1
paraboloid
17
0
I'm given this definite integral:
[itex]\int_0^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)dzdydx[/itex]

I need to change the order to dydxdz, but I'm stuck trying to get the limits of integration wrt y.

24on636.png


[itex]\int_0^{1}\int_{x^2}^{0}\int_{}^{}f(x,y,z)dydzdx[/itex]

How do I find the limits of y?
 
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  • #2
Just look at the original iterated integral to see the region over integration occurs.

z = 0 to z = 1 - y
y = sqrt(x) to y = 1
x = 0 to x = 1.

Sketch that region and it will help you determine the limits of integration for a different order of integration.
 

What is the purpose of changing the order of a triple integration?

The purpose of changing the order of a triple integration is to simplify the integration process and make it easier to solve. By changing the order, the integrand may become easier to integrate, potentially saving time and effort.

How do you change the order of a triple integration?

To change the order of a triple integration, you can rearrange the integrals in any order that makes it easier to solve. This can be done by using properties of integration such as Fubini's theorem or by making use of symmetry in the integrand.

What are the benefits of changing the order of a triple integration?

Changing the order of a triple integration can make the integration process more manageable and potentially save time and effort. It can also help to reveal patterns and relationships between the variables in the integrand, providing a deeper understanding of the problem at hand.

Are there any limitations to changing the order of a triple integration?

There may be limitations to changing the order of a triple integration, particularly when dealing with complex functions or regions of integration. It is important to carefully consider the limits of integration when changing the order to ensure the correct solution is obtained.

Can the order of a triple integration be changed for any type of function?

The order of a triple integration can be changed for most types of functions, including polynomials, trigonometric functions, and exponential functions. However, there may be cases where changing the order is not possible or not advantageous, such as when dealing with discontinuous functions or non-rectangular regions of integration.

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