Find the volume of an object set with the following function: x+y+z=1.

Nina Bean

Nina Bean

Answered question

2022-08-13

Volume integral - finding the region
I need to find the volume of an object set with the following function:
x + y + z = 1
And all three axis.
So I converted the function into z = x y + 1, and it gave me kind of a clepsydra, crossing x,y plane in x = y = 1.
So the region seems to be the circle x 2 + y 2 = 1.
However, changing x to r, and y to ϕ gives me wrong result π, and it should be 1/6.
Is the region I've came up with correct?

Answer & Explanation

prelatiuvq

prelatiuvq

Beginner2022-08-14Added 6 answers

Explanation:
V ( 0 , 1 ) 3 [ x + y + z < 1 ] d x d y d z =   0 1 0 1 0 1 [ x < 1 y z ] d x d y d z =   0 1 0 1 [ 0 < 1 y z < 1 ] 0 1 y z d x d y d z =   0 1 0 1 [ z < y < 1 z ] ( 1 y z ) d y =   0 1 0 1 [ y < 1 z ] ( 1 y z ) d y d z =   0 1 0 1 z ( 1 y z ) d y d z =   1 2 0 1 ( 1 z ) 2 d z = 1 6
logosdepmpe

logosdepmpe

Beginner2022-08-15Added 4 answers

Step 1
First of all, the region of interest here is a right tetrahedron, with a right triangle base of side length 1 and height 1, so the volume is 1 3 1 2 ( 1 ) ( 1 ) = 1 6 .
The volume integral may be done by integrating over, say, x first, then y, then z:
V = 0 1 d z 0 1 z d y 0 1 y z d x
Step 2
Doing the innermost integral:
V = 0 1 d z 0 1 z d y ( 1 y z ) = 0 1 d z [ ( 1 z ) 2 1 2 ( 1 z ) 2 ] = 1 2 0 1 d z ( 1 z ) 2
Doing the final integral, V = 1 2 1 3 1 = 1 6

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school geometry

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?