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Calculus IIIChapter 4: Multiple IntegralsSection 4-5

Triple Integrals — Question 5

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Question 5

Find the average of f=x2+y2+z2f=x^2+y^2+z^2 on [−1,1]3[-1,1]^3.

Tasks

  1. Use symmetry and separability.

  2. Compute the average.

  3. Compare with the range.

Original worksheet page 1: question and worked solution for 4-5-005
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Question 5 – Solution

Strategy. The three squared-coordinate contributions are identical.

Step 1: Integral∭Bx2dV=(∫−11x2dx)(2)(2)=8/3\iiint_Bx^2dV=(\int_{-1}^1x^2dx)(2)(2)=8/3. Thus ∭BfdV=3(8/3)=8\iiint_Bf\,dV=3(8/3)=8.

Step 2: AverageSince vol⁡(B)=8\operatorname{vol}(B)=8, favg=1\boxed{f_{\mathrm{avg}}=1}.

Verification0≤f≤30\le f\le 3, and each coordinate square has average 1/31/3, so their sum has average 11.

Original worksheet page 2: question and worked solution for 4-5-005

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