Integrating f(x,y,z): Confirm Ranges for x, y, z

In summary, the problem involves finding the triple integral of x^2 over the tetrahedron G bounded by the coordinate planes and the plane octant x + y + z = 1. The ranges for x, y, and z are correctly set up as 0 to 1 for z, 0 to 1-x for y, and 0 to 1 for x. The incorrect answer was due to an integration error, which has since been resolved.
  • #1
naspek
181
0

Homework Statement



f(x, y, z) = x^2 ; G is tetrahedron bounded by the coordinate planes and the plane octant with equation x + y + z = 1

∫ ∫ ∫ x^2 dzdydx

I try to set up the ranges for x, y and z..

x+y+z = 1
z = 1-x-y...set the limits for z from z=0 to z = 1-x-y

x+y+z = 1
if, z = 0, y = 1-x ...set the limits for y = 0 to y = 1-x

x+y+z = 1
if, z = 0 , y = 0 ...x = a set the limits from x = 0 to x =1

first.. i need help to confirm that my ranges are correct..
because.. I've done the integration and got the wrong answer..
i've double checked my integration and it is right..
there must be something wrong with my ranges.. i guess...
 
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  • #2
They look right to me. Why don't you show what you did. It may be an integration error.
 
  • #3
yup2.. it's integration error.. duuhh...~
thanks btw..
problem solved..
 

1. What is the purpose of integrating f(x,y,z) and what does it represent?

The purpose of integrating f(x,y,z) is to calculate the total value of a function over a specified range. This represents the volume under the surface defined by the function in three-dimensional space.

2. How do you determine the ranges for x, y, and z when integrating f(x,y,z)?

The ranges for x, y, and z are typically determined by the boundaries of the region of interest in three-dimensional space. These boundaries can be determined by looking at the context of the problem or by graphing the function in question.

3. Can the ranges for x, y, and z be negative when integrating f(x,y,z)?

Yes, the ranges for x, y, and z can be negative when integrating f(x,y,z). This is especially common when dealing with symmetric functions or functions that have negative values in the given range.

4. How do you confirm the ranges for x, y, and z when integrating f(x,y,z)?

To confirm the ranges for x, y, and z, you can double check the boundaries of the region of interest and ensure that they are consistent with the limits of integration in the given problem. You can also plot the function in question to visually confirm the ranges.

5. Are there any special cases to consider when confirming the ranges for x, y, and z when integrating f(x,y,z)?

Yes, there are some special cases to consider when confirming the ranges for x, y, and z. For example, if the function being integrated is undefined or has singularities at certain points, these points must be excluded from the range. Additionally, if the function is discontinuous, the ranges may need to be broken up into smaller intervals for more accurate integration.

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