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Calculus IIIChapter 4: Multiple IntegralsSection 4-5

Triple Integrals — Question 8

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Question 8

Let I(a)=∫01∫02∫03(a+x−y+z)dzdydxI(a)=\int_0^1\int_0^2\int_0^3(a+x-y+z)\,dz\,dy\,dx. Find aa so the average integrand is 55.

Tasks

  1. Evaluate I(a)I(a).

  2. Solve the average condition.

  3. Verify using coordinate averages.

Original worksheet page 1: question and worked solution for 4-5-008
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Question 8 – Solution

Strategy. Convert average 55 into total 55 times box volume.

Step 1: IntegralThe volume is 66 and the coordinate averages are (1/2,1,3/2)(1/2,1,3/2), hence I(a)=6(a+1)I(a)=6(a+1).

Step 2: SolveI(a)/6=5I(a)/6=5 gives a+1=5a+1=5, so a=4\boxed{a=4}.

VerificationAt a=4a=4, the average is 4+1/2−1+3/2=54+1/2-1+3/2=5, and the total integral is 3030.

Original worksheet page 2: question and worked solution for 4-5-008

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