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Vector calculus- region-density-mass

  • #1

Homework Statement


View attachment 229290
upload_2018-8-15_16-23-12.png

Homework Equations



upload_2018-8-15_16-24-25.png

The Attempt at a Solution


39245723_483901692074113_9059923629021593600_n.jpg

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

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Answers and Replies

  • #2
Chandra Prayaga
Science Advisor
649
148
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.
 
  • #3
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.
39177607_1094553904033685_3707489651934625792_n.jpg

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?
 

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Last edited:
  • #5
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
32,781
5,050
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?
 

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