# How to determine the limits for triple integration?

## 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
Staff Emeritus
Science Advisor
Homework Helper

## 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
Science Advisor
Homework Helper
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.

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
Staff Emeritus
Science Advisor
Homework Helper
In cases like this, making a simple sketch can often provide clarity to what the limits are.

• Alex_Neof
HallsofIvy
Science Advisor
Homework Helper
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?

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.