# Thread: Switching the order of triple integrals

1. ## Switching the order of triple integrals

Well this is just a nightmare and I got my first C on an exam in college after a year, going to finish calculus III with triple integrals and I can't figure this out at all

the region is:

Q: { (x,y,z) : x: [0,1] y: [0,x] and z: [0,3] }

How do I find the boundaries of z? I can set up the integrals dzdxdy, dzdydx, I can also draw the region, I get the idea in 2d but I don't get how the bounds of integration for x are from 1 to y. How??

2. ## Re: Switching the order of triple integrals

You are told that the "boundaries on z" are 0 and 3!

Ignoring z, you are told that x goes from 0 to 1 and that, for each x, y goes from 0 to x. On a Cartesian coordinate system, draw vertical lines at x= 0 and x= 1 marking those boundaries. y= 0 is a third boundary and y= x, the fourth boundary goes from (0, 0) to (1, 1). Since y goes up from 0 to x, the area of integration is the triangle below y= x. To reverse the order of integration, observe that, over all, y goes from 0 to 1 and that, for every y, x goes from the line x= y to 1.

$\displaystyle \int_{z= 0}^3\int_{x= 0}^1\int_{y= 0}^x f(x,y,z)dydxdz$
$\displaystyle \int_{z= 0}^3\int_{y= 0}^1\int_{x= y}^1 f(x,y,z)dxdydz$
$\displaystyle \int_{x= 0}^1\int_{z= 0}^3\int_{y= 0}^x f(x,y,z)dydzdx$
$\displaystyle \int_{x= 0}^1\int_{y= 0}^x\int_{z= 0}^3 f(x,y,z)dzdydx$
$\displaystyle \int_{y= 0}^1\int_{x= y}^1\int_{z= 0}^3 f(x,y,z)dzdxdy$
$\displaystyle \int_{y= 0}^1\int_{z= 0}^3\int_{x= y}^1 f(x,y,z)dxdzdy$

3. ## Re: Switching the order of triple integrals

Thank you so much! That was extremely helpful.