Triple integrals, changing the order of integration

In summary, the triple integral for the volume of the solid shown can be written as \displaystyle \int_{0}^{5} \int_{0}^{1 - 4z/5} \int_{0}^{1 - 2z/5\,} dx\,dy\,dz. Two possible orders of integration are from the innermost to outermost as x, y, z and vice versa, with limits of integration as described above.
  • #1
amalone
2
0

Homework Statement



Write out the triple integral for the volume of the solid shown in all six possible orders. Evaluate at least 2 of these integrals.


Homework Equations



I attached a picture of the figure. The front : x/2+z/5=1
right : y/4+z/5=1

The Attempt at a Solution



I really need help with the possible orders and not the actual integration.
I figured out two different orders but I'm not sure how to post them properly
 

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  • #2
amalone said:

Homework Statement



Write out the triple integral for the volume of the solid shown in all six possible orders. Evaluate at least 2 of these integrals.

Homework Equations



I attached a picture of the figure. The front : x/2+z/5=1
right : y/4+z/5=1

The Attempt at a Solution



I really need help with the possible orders and not the actual integration.
I figured out two different orders but I'm not sure how to post them properly
Hello amalone. Welcome to PF !

attachment.php?attachmentid=55246&d=1359687598.jpg


You can simply describe the order in which you do the integrations, along with the limits of integration.

For example:
(From inner to outer)
Integrate over x from x = 0 to x = 1 - 2z/5 .

Integrate over y from y = 0 to y = 1 - 4z/5 .

Integrate over z from z = 0 to z = 5 .​

You could learn LaTeX and write:

[itex]\displaystyle \int_{0}^{5} \int_{0}^{1 - 4z/5} \int_{0}^{1 - 2z/5\,} dx\,dy\,dz[/itex]
 
  • #3
SammyS said:
Hello amalone. Welcome to PF !

attachment.php?attachmentid=55246&d=1359687598.jpg


You can simply describe the order in which you do the integrations, along with the limits of integration.

For example:
(From inner to outer)
Integrate over x from x = 0 to x = 1 - 2z/5 .

Integrate over y from y = 0 to y = 1 - 4z/5 .

Integrate over z from z = 0 to z = 5 .​

You could learn LaTeX and write:

[itex]\displaystyle \int_{0}^{5} \int_{0}^{1 - 4z/5} \int_{0}^{1 - 2z/5\,} dx\,dy\,dz[/itex]

Thanks!
 
  • #4
How about describing the orders of integration that you've found ?
 

FAQ: Triple integrals, changing the order of integration

What is a triple integral and when is it used?

A triple integral is a mathematical concept used to calculate the volume of a three-dimensional region. It is typically used in multivariable calculus and is an extension of the concept of a double integral.

How do you change the order of integration for a triple integral?

To change the order of integration for a triple integral, you must first determine the limits of integration for each variable. Then, you can rearrange the order of the variables in the integral and adjust the limits accordingly. This is often done to simplify the integral and make it easier to solve.

What is the significance of changing the order of integration in a triple integral?

Changing the order of integration in a triple integral can make the integral easier to solve or may reveal symmetries that can simplify the calculation. It can also help in visualizing the region being integrated over and understanding the geometry of the problem.

Can the order of integration be changed for any triple integral?

Yes, the order of integration can be changed for any triple integral as long as the limits of integration can be properly adjusted. However, some integrals may be more difficult to solve in certain orders, so it is important to carefully choose the most efficient order for a specific problem.

Are there any specific rules or techniques for changing the order of integration in triple integrals?

Yes, there are certain rules and techniques that can be used to change the order of integration in triple integrals. These include using symmetry, making appropriate variable substitutions, and using geometric intuition to determine the most efficient order. It is also important to carefully check the limits of integration after changing the order to ensure they are still correct.

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