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Calculus IIIChapter 4: Multiple IntegralsSection 4-5

Triple Integrals — Question 4

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Question 4

A box 0≤x≤20\le x\le 2, 0≤y≤10\le y\le 1, 0≤z≤30\le z\le 3 has density ρ=1+x+z\rho=1+x+z.

Tasks

  1. Find its mass.

  2. Find the average density.

  3. Check against the density range.

Original worksheet page 1: question and worked solution for 4-5-004
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Question 4 – Solution

Strategy. Integrate density and divide by volume.

Step 1: MassM=∫02∫01∫03(1+x+z)dzdydx=∫02(15/2+3x)dx=21M=\int_0^2\int_0^1\int_0^3(1+x+z)dz\,dy\,dx=\int_0^2(15/2+3x)dx=\boxed{21}.

Step 2: AverageThe volume is 66, so ρavg=21/6=7/2\boxed{\rho_{\mathrm{avg}}=21/6=7/2}.

VerificationHere 1≤ρ≤61\le\rho\le 6; 7/27/2 lies in this interval. Also the coordinate-average rule gives 1+1+3/2=7/21+1+3/2=7/2.

Original worksheet page 2: question and worked solution for 4-5-004

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