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May 28th, 2017, 01:51 PM   #1
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Calculating density with strange shape...

Calculate the mass of solid V bounded by given planes and having density p(x,y,z) = 3 + 2x + 2y - 2z

x = 0, y = 0, z = 0, x + y = 1, z = x + 2y

Answer: 1/6
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May 28th, 2017, 04:03 PM   #2
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I managed to get the following boundaries...


+


= 1/6

Can anyone confirm the boundaries for me? Specially the first term y lower bound 1 - z?

Last edited by zollen; May 28th, 2017 at 04:06 PM.
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May 29th, 2017, 03:11 AM   #3
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The order of your integration is awkward.

We are given x = 0, y = 0, z = 0, x + y = 1, z = x + 2y.

The first three say that this object is bounded by the coordinate planes- we are in the first octant. x+ y= 1 gives a boundary in the xy- plane which then extends vertically. The top is z= x+ 2y.

The line x+ y= 1 goes from (1, 0) to (0, 1) so x varies from 0 to 1. For every x, y varies from 0 to 1- x. Finally, for every (x, y), z varies from 0 to x+ 2y. The integral for mass is $\displaystyle \int_0^1\int_0^{1- x}\int_0^{x+ 2y} (3 + 2x + 2y - 2z) dzdydx$

(And you are calculating the mass, not the density.)

Last edited by Country Boy; May 29th, 2017 at 03:14 AM.
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