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Calculus IIIChapter 4: Multiple IntegralsSection 4-5

Triple Integrals — Question 9

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Question 9

For continuous f,g,hf,g,h on their respective intervals, prove the product rule for f(x)g(y)h(z)f(x)g(y)h(z) on [a,b]×[c,d]×[e,k][a,b]\times[c,d]\times[e,k], then apply it to (1+x)y2ez(1+x)y^2e^z on [0,1]×[−1,1]×[0,ln⁡2][0,1]\times[-1,1]\times[0,\ln 2].

Tasks

  1. Prove factorization.

  2. Evaluate the example.

  3. Check the sign.

Original worksheet page 1: question and worked solution for 4-5-009
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Question 9 – Solution

Strategy. Pull factors independent of the current integration variable outside each inner integral.

Step 1: ProofSuccessive integration gives ∭fghdV=(∫abf)(∫cdg)(∫ekh)\iiint fgh\,dV=(\int_a^bf)(\int_c^dg)(\int_e^kh). Continuity guarantees the orders are valid.

Step 2: Example∫01(1+x)dx=3/2\int_0^1(1+x)dx=3/2, ∫−11y2dy=2/3\int_{-1}^1y^2dy=2/3, and ∫0ln⁡2ezdz=1\int_0^{\ln 2}e^zdz=1. Their product is 1\boxed{1}.

VerificationAll factors are nonnegative on the box, matching the positive result.

Original worksheet page 2: question and worked solution for 4-5-009

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