Vector calculus- region-density-mass

[Jozefina Gramatikova]

Homework Statement

https://www.physicsforums.com/attachments/229290

Homework Equations

The Attempt at a Solution

I am not sure what equation to use for the volume[/B]

 

[Chandra Prayaga]

The attempt that you posted is too small and I cannot click on it to get a better view. But from what I can see, I suggest the following.
1. Understand, from the ranges of x, y and z given by you, the shape of the region you are interested in. In other words, answer the first question and sketch the region. In your graph, you only showed a few isolated points. You did not sketch the region in space.
2. The relevant equation posted by you is not adequate. Density = mass / volume is good only if the density is constant. In your case, the density depends on position (your equation: ρ = 1 + z). Then you must use the differential form:

dm = ρ dV,

and integrate this to get the total mass. It will be a triple integral.

 

[Jozefina Gramatikova]

The attempt that you posted is too small and I cannot click on it to get a better view. But from what I can see, I suggest the following.
1. Understand, from the ranges of x, y and z given by you, the shape of the region you are interested in. In other words, answer the first question and sketch the region. In your graph, you only showed a few isolated points. You did not sketch the region in space.
2. The relevant equation posted by you is not adequate. Density = mass / volume is good only if the density is constant. In your case, the density depends on position (your equation: ρ = 1 + z). Then you must use the differential form:

dm = ρ dV,

and integrate this to get the total mass. It will be a triple integral.

I hope you can see the picture better now. I was wondering if I need the whole rectangle as a region or just the purple triangle that I sketched there?

 

[Jozefina Gramatikova]

I guess it is just the triangle, because we have

Is my final equation ok?

 

[haruspex]

I was wondering if I need the whole rectangle as a region or just the purple triangle that I sketched there?

Your sketch is nowhere close to being right. It is a three dimensional solid.
What is the range of z? For some arbitrary z in that range, what does the XY lamina look like?