How to determine the limits for triple integration?

[Alex_Neof]

Homework Statement

Evaluate the triple integral:
∫ x dxdydz
A

where

A = {(x; y; z) : x, y, z > 0, x + y + z ≤ 1} .

Homework Equations

None that I know of.

The Attempt at a Solution

The problem I have is determining the limits for x, y and z. I don't really understand the following notation:

A = {(x; y; z) : x, y, z > 0, x + y + z ≤ 1} , which I believe will help me.

I'm guessing x + y + z ≤ 1 is the definition of a plane and the region A that we are integration over is in the first octant since x, y, z > 0.

Kind regards.

 

[SteamKing]

Homework Statement

Evaluate the triple integral:
∫ x dxdydz
A

where

A = {(x; y; z) : x, y, z > 0, x + y + z ≤ 1} .

Homework Equations

None that I know of.

The Attempt at a Solution

The problem I have is determining the limits for x, y and z. I don't really understand the following notation:

A = {(x; y; z) : x, y, z > 0, x + y + z ≤ 1} , which I believe will help me.

I'm guessing x + y + z ≤ 1 is the definition of a plane and the region A that we are integration over is in the first octant since x, y, z > 0.

Kind regards.

You are correct.

However, in future, please post your homework in the proper HW forum. I'm moving this thread to the Calculus HW forum.

 

[HallsofIvy]

x+ y+ z= 1, which is the same as z= 1- x- y, is a plane crossing the three coordinate axes at (1, 0, 0), (0, 1, 0), and (0, 0, 1).

Projecting down to the xy-plane, where z= 0, we have the region bounded by the x and y axes and the line x+ y= 0. The line x+ y= 0, which is the same as y= 1- x, crosses the x and y axes at (1, 0) and (0, 1). So x goes from a smallest value of 0 to a largest value of 1. Now, if you were to draw a line parallel to the y-axis, for each x, what would the lowest and largest values of y be? If you were to draw a line parallel to the z- axis, for each (x,y), what would the smallest and largest values of y be?

 

[Alex_Neof]

Ah sorry about that SteamKing. So if I consider layers along the z axis, my constant limits for z would be from 0 to 1. I still don't know how to determine the limits for x and y. I cannot visualise the projection of A onto the x-y plane.

 

[Alex_Neof]

Hi HallsofIvy, so for each value of x, the lowest and highest value of y would be from 0 to the line x+y=1, so the highest value will be y=x-1? Will the highest values for z be 0 to 1?

Kind regards.

ah sorry, I think my limits for z are wrong as you told me to think about z for each (x,y)

 

[SteamKing]

In cases like this, making a simple sketch can often provide clarity to what the limits are.

 

[HallsofIvy]

Hi HallsofIvy, so for each value of x, the lowest and highest value of y would be from 0 to the line x+y=1, so the highest value will be y=x-1?

No! If x+ y= 1 then y= 1- x.

Will the highest values for z be 0 to 1?

No.

Kind regards.

ah sorry, I think my limits for z are wrong as you told me to think about z for each (x,y)

z goes from 0 to 1- x- y.

 

[Alex_Neof]

HallsofIvy, why does z go from 0 to 1-x-y?

 

[Alex_Neof]

ah it's the equation of the plane from the xy plane to z=1- x -y for each x and y! nevermind!
Thank you! I really appreciate the help from everyone.