Triple Integrals — Question 2

Question 2

Let $E=\{0\le x\le 1,\ 0\le y\le 1-x,\ 0\le z\le 2-x-y\}$.

Tasks

1. Interpret the bounds.

2. Evaluate $\iiint_E1\,dV$.

3. Check positivity of every bound.

Question 2 – Solution

Strategy. Integrate vertical height over the triangular projection.

Step 1: GeometryThe projection is the triangle $x\ge 0$, $y\ge 0$, $x+y\le 1$. Above it, the solid runs from $z=0$ to the sloping plane $z=2-x-y$.

Step 2: Evaluate$V=\int_0^1\int_0^{1-x}\int_0^{2-x-y}1\,dz\,dy\,dx=\int_0^1\left(\frac 32-2x+\frac{x^2}{2}\right)dx=\boxed{\frac 23}$.

VerificationOn the projection, $2-x-y\ge 1$, so every vertical bound is positive.