Another Triple Integral Question

In summary, the conversation is about understanding the limits of integration for a solid bounded by a parabolic cylinder and three planes. The question is how to visualize the solid and determine the limits of integration, as well as the order of integration. The link provided is not working and the person offering help suggests using Latex coding. The expert then provides a solution using divide and conquer method. The conversation concludes with the person understanding the solution and thanking the expert for their help.
  • #1
Theelectricchild
260
0
Hello.

Here is the original question http://

My difficuly is understanding the limits of integration--- party due to how the solid in question is "sliced" by those planes. I know how to visualize the parabolic cylinder, but I need help on 1. limits on integration, and 2. Order of integration.

I doubt I would have to use polar coordinates since the region in question has no square roots...

Thanks for you help I really do appreciate this.
 
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  • #2
Arghh that links not working, just copy paste it into the url. Sorry.
 
  • #3
TEChild, you can use Latex coding here, click on this link:
https://www.physicsforums.com/showthread.php?t=8997

[tex]\int \int \int_E (x + 2y)dV[/tex]

where E is bound by the parabolic cylinder

[tex]y=x^2[/tex]

and the planes

[tex]x=z,x=y,z=0[/tex]
 
Last edited:
  • #4
Divide and conquer. z appears as an independent variable once and as a constant. So, easy to eliminate x:

[tex]\int _E ()dV = \int_0^1 dz \int dy \int_z^1 ()dx[/tex]

Do you see why the upper limit on z is 1? Solve the innermost integral as tho y and z were constants.
 
  • #5
actually I don't see why the upper limit on z is one... that's where I was confused --- I understand why it starts at 0 of course... and also the y limits are giving me trouble...
 
  • #6
taking z as the independent variable, ask yourself: what is the greatest z value a point in the bounded volume can have?
 
  • #7
ahhhh i got it thanks so much
 
  • #8
Ooh Bellingham--- are you a graduate student at Western Washington U. outandbeyond?
 

1. What is a triple integral?

A triple integral is a mathematical concept used to calculate the volume of a three-dimensional object. It involves integrating a function over a three-dimensional region. The result of a triple integral is a numerical value that represents the volume of the object.

2. How do you set up a triple integral?

To set up a triple integral, you need to determine the limits of integration for each variable (x, y, z) and the function to be integrated. This can be done by visualizing the three-dimensional region and identifying the boundary surfaces. The order of integration (which variable is integrated first, second, and third) is also important and can be determined by the shape of the region.

3. What are some real-world applications of triple integrals?

Triple integrals have various applications in science and engineering. They are commonly used in physics to calculate the mass and center of mass of a three-dimensional object. They also have applications in fluid mechanics for calculating fluid flow rates and in calculus to find the moment of inertia of an object.

4. What are some techniques for solving triple integrals?

There are various techniques for solving triple integrals, including using geometric properties of the region, changing the order of integration, and using symmetry. Another common technique is to convert the triple integral into a double integral by using the method of cylindrical or spherical coordinates.

5. What are some common mistakes made when solving triple integrals?

Some common mistakes when solving triple integrals include incorrectly setting up the limits of integration, using the wrong order of integration, and making computational errors. It is also important to pay attention to the boundaries and make sure they are correctly identified and integrated over.

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