Finding Volume in Positive Octant using Multiple Integration

[AlmostFamous]

okay, I'm not as bad at this as all the questions make me out to be lol

here's the question I've got...

Find the volume bound by 2x+z=3 and y+3z=9 in the positive octant, i.e. x, y and z >=0

what i tried was finding the limits on y first. i got from 0 to 9-3z
for x : 0 to $\frac{z-3}\2$

but then for z i can't find any.

saying that, i don't think the first two are right. I've always had trouble deciding the limits for a 3D object. Any help on the limits would be appreciated.

 

[HallsofIvy]

What is your order of integration? Since you have limits of integration for x and y that depend on z, I assume that z is the "outer integral". In that case, the limits of integration, for z, must be constant. It's clear that the lower limit is z= 0. What is the largest possible value for z?

However, I'm not certain that the single integral will give the correct value. It looks to me like you are going to need to break this into two integrals at the place where the two planes intersect.

 

[AlmostFamous]

There isn't an order specified, but I would also assume that z would be the outer integral.

Am i right in saying that the largest value for z would be 3, so the z limits are 0 to 3?

 

[AlmostFamous]

i've looked at this question again, and I don't know what you mean by splitting it up. I've never had a question like that before.

 

[GCT]

first integrate from 0 to y=9-3z with respect to y, then 0 to 3-2x with respect to z, then find numerical boundaries of the domain with respect to x, from zero to...you should be able to figure it out. All of the info is self contained, one is covering all of the volume elements of the specified boundary. The function to integrate is simply 1. Kinda remember this from multivariable which I took this past semester.