Triple Integrals — Question 1

Question 1 – Solution

Strategy. Integrate termwise over the box.

Step 1: Set up$I=\int_0^1\int_0^2\int_{-1}^1(x+2y+3z)\,dz\,dy\,dx$.

Step 2: EvaluateThe $z$ term is odd; the other terms give $2x+4y$. Thus $I=\int_0^1\int_0^2(2x+4y)dy\,dx=\int_0^1(4x+8)dx=\boxed{10}$.

VerificationThe box volume is $4$ and its coordinate averages are $(1/2,1,0)$, so the integrand average is $5/2$ and $4(5/2)=10$.