Volume and Mass of an object using integration

[theno1katzman]

Homework Statement

Find exactly the volume in the first octant under the planes x + z =1 and y + z =1. Find the mass if the density is p(x,y,z)=xyz.

The Attempt at a Solution

This is a pyramid in shape, so for the volume, rather than constructing a double integral, I used the formula V= Bh/3 which is equal to 1/3.

Please see the jpg file for the work on finding the mass.
Where did I go wrong? It came out to be 0.
I tried reversing the z limits and that didn't do anything.

 

[LCKurtz]

You are integrating in the z direction first. But z on the upper surface is not a single formula. The "roof" of your pyramid has two pieces so you must break it up into two integrals. This will also affect your limits in the middle integral. Also, what is z on the "floor"?

 

[theno1katzman]

Thanks here is a second attempt at the problem, please let me know if what I did here is right.

 

[LCKurtz]

No, it isn't. You are integrating over the xy square twice. You need a different "floor" area for each section of the "roof".

 

[BatmanACC]

Homework Statement

Find exactly the volume in the first octant under the planes x + z =1 and y + z =1. Find the mass if the density is p(x,y,z)=xyz.

The Attempt at a Solution

This is a pyramid in shape, so for the volume, rather than constructing a double integral, I used the formula V= Bh/3 which is equal to 1/3.

Please see the jpg file for the work on finding the mass.
Where did I go wrong? It came out to be 0.
I tried reversing the z limits and that didn't do anything.

Hey mate, I don't have time to run over the specifics right now (studying for my own exams!) but here's how this is done.

Edit...I just told you how to calculate the mass of a wire LOL wow it's getting late, let me re-do this

Okay note that the intersection of these curves happens at y=x which is a straight line across the XY plane. Hmmm...this result tells us something.

It tells us that to remain in the first octant z must be less then 1 or else we will have a negative x or a negative y (which are out of the first quadrent). So we have a bound for z, 0-->1

So what are our y and x bounds then? Well we know that x runs from 0-->1-z as does y. Well we can now express this as a double integral without problems,

integrate (x^2 * z) using good bounds and you're home free. Note that x = y is subbed in here, alternatively you could keep it at xyz. Integrate x and y first because they depend on Z and your Z integral then has coefficient bounds. Hurray!

That should be it I'm a little flustered as you can tell (at first I thought you were asking the mass of a wire about those curves LOL!)

 

[theno1katzman]

Here is the solution that uses the base areas properly. I drew a diagram for this area as well. Arrows point to surfaces and the surfaces are using drawn with arrows pointing to the pyramids apex.
m=1/120
I hope I have it now. This problem has been bugging me for some time ha-ha.