To evaluate the triple integral

$\iiint_E xy \, dV$

where $E$ is the solid tetrahedron with vertices at $(0,0,0)$, $(7,0,0)$, $(0,6,0)$, and $(0,0,8)$, we need to:

1. Define the region $E$ in terms of the boundaries for $x$, $y$, and $z$.
2. Set up the limits for the triple integral based on this region.
3. Integrate $xy$ over $E$.

Step 1: Determine the Boundaries of $E$

Since $E$ is a tetrahedron defined by the vertices $(0,0,0)$, $(7,0,0)$, $(0,6,0)$, and $(0,0,8)$, each vertex provides a boundary constraint for $x$, $y$, and $z$.

The plane containing these points forms a triangular boundary:

1. From $(0,0,0)$ to $(7,0,0)$, $x$ ranges from $0$ to $7$.
2. From $(0,0,0)$ to $(0,6,0)$, $y$ ranges from $0$ to $6$ but depends on $x$.
3. From $(0,0,0)$ to $(0,0,8)$, $z$ ranges from $0$ to $8$ but depends on both $x$ and $y$.

The equations of the planes of the tetrahedron can be deduced by solving the boundaries:

• From $x = 7 - \frac{7}{6}y$ for the limits of $y$,
• Then