Changing order of integration

[jwpaine]

Hey All!

I have the triple integral:

\(\displaystyle \int_0^1\int_{1-x}^1\int_x^1 f(x,y,z)dzdydx\)

I am asked to sketch the region of integration: I assume this would be the lines z = x, z = 1, y = 1-x, y=1, and x = 1, correct?

I am then asked to express the integral with order of integration dxdydz AND then express with order of integration dydxdz

How do I go about setting up my limits based on the change in order? I assume it's not as easy as simply, changing the order of integration and that I'll have to use my graphed region as reference?

Help would be great!

 

[Deleted member 4993]

Hey All!

I have the triple integral:

\(\displaystyle \int_0^1\int_{1-x}^1\int_x^1 f(x,y,z)dzdydx\)

I am asked to sketch the region of integration: I assume this would be the lines z = x, z = 1, y = 1-x, y=1, and x = 1, correct?

I am then asked to express the integral with order of integration dxdydz AND then express with order of integration dydxdz

How do I go about setting up my limits based on the change in order? I assume it's not as easy as simply, changing the order of integration and that I'll have to use my graphed region as reference?

Help would be great!

First you need to figure out - what does your region look like:

z=x is a plane parallel to y-axis. And similarly y = 1-x is a plane parallel to z-axis.

Sketch those along with the other planes. It will pop-up.

 

[jwpaine]

Yes, that was stupid of me. I'll post my solution so if someone uses the search function and finds this problem, they'll have a workable solution:

For the order of integration dxdydz
y = 1-x => x = 1-y
z = x
z = x = 1-y
y = 1 - z


0 <= x <= 1-y
1-z <= y <= 1
0 <= z <= 1

Which gives us the integral \(\displaystyle \int_0^1\int_{1-z}^1\int_0^{1-y} f(x,y,z)dxdydz\)

For the order of integration dydxdz:
for z = x, z = 1, y = 1-x, y = 1, x=0, x=1
1-x <= y <= 1
0 <= x <= z
0 <= z <= 1

Giving us the integral \(\displaystyle \int_0^1\int_0^z\int_{1-x}^1 f(x,y,z)dydxdz\)

Cheers!
John