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 May 28th, 2017, 01:51 PM #1 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 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
 May 28th, 2017, 04:03 PM #2 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 I managed to get the following boundaries... $ \int_{0}^{1} \int_{1-z}^{1} \int_{2(z-y)}^{1-y} (3+2x+2y-2z) dxdydz $ + $ \int_{0}^{1} \int_{z}^{1} \int_{z-2y}^{1-y} (3+2x+2y-2z) dxdydz $ = 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.
 May 29th, 2017, 03:11 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 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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