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Calculus IIIChapter 4: Multiple IntegralsSection 4-5

Triple Integrals — Question 10

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Question 10

A solid EE has volume 1212, and 2≤f≤72\le f\le 7 throughout EE. Another integrable function gg satisfies |g−f|≤0.05|g-f|\le 0.05. Given ∭EfdV=50\iiint_Ef\,dV=50, bound ∭EgdV\iiint_Eg\,dV.

Tasks

  1. Check consistency of the data.

  2. Derive sharp bounds for the new integral.

  3. Explain sharpness.

Original worksheet page 1: question and worked solution for 4-5-010
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Question 10 – Solution

Strategy. Integrate pointwise inequalities and scale uniform error by volume.

Step 1: Consistency24≤∭EfdV≤8424\le\iiint_Ef\,dV\le 84, so 5050 is possible.

Step 2: PerturbationFrom −0.05≤g−f≤0.05-0.05\le g-f\le 0.05 and vol⁡(E)=12\operatorname{vol}(E)=12, −0.6≤∭E(g−f)dV≤0.6.-0.6\le\iiint_E(g-f)dV\le 0.6. Therefore 49.4≤∭EgdV≤50.6\boxed{49.4\le\iiint_Eg\,dV\le 50.6}.

VerificationThe endpoints occur for g=f−0.05g=f-0.05 and g=f+0.05g=f+0.05, so the supplied data guarantee no tighter interval.

Original worksheet page 2: question and worked solution for 4-5-010

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