Triple Integrals — Question 3

Question 3

Find the volume of the first-octant tetrahedron $x/2+y/3+z/6\le 1$.

Tasks

1. Write bounds in the order $dz\,dy\,dx$.

2. Evaluate.

3. Verify geometrically.

Question 3 – Solution

Strategy. Solve the plane successively for $z$ and for the projected $y$-boundary.

Step 1: GeometryThe coordinate-plane faces meet the sloping face at the three axis intercepts shown.

Step 2: Bounds and integral$0\le x\le 2$, $0\le y\le 3-3x/2$, and $0\le z\le 6-3x-2y$. Hence $V=\int_0^2\int_0^{3-3x/2}(6-3x-2y)dy\,dx=\int_0^2\frac 94(2-x)^2dx=\boxed{6}$.

VerificationThe intercept formula gives $V=(1/6)(2)(3)(6)=6$.