Triple Integrals — Question 7

Question 7 – Solution

Strategy. Each order peels coordinates from the simplex inequality.

Step 1: GeometryThe simplex is bounded by the coordinate planes and $x+y+z=1$. The two marked fibers show which coordinate is peeled off first.

Step 2: Orders$I=\int_0^1\int_0^{1-x}\int_0^{1-x-y}z\,dz\,dy\,dx=\int_0^1\int_0^{1-y}\int_0^{1-y-z}z\,dx\,dz\,dy$. The first form gives $I=\frac 16\int_0^1(1-x)^3dx=\boxed{\frac 1{24}}$.

VerificationThe simplex volume is $1/6$ and its centroid has $\bar z=1/4$, so the moment is $(1/6)(1/4)=1/24$.