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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

Evaluate the triple integral ∭_E xy dV where E is the solid tetrahedron with vertices (0,0,0), (7,0,0), (0,6,0), (0,0,8).

Calculate the triple integral of xy over the volume of a tetrahedron defined by vertices (0,0,0), (7,0,0), (0,6,0), and (0,0,8). This problem involves setting up a multivariable integral, finding the region boundaries, and integrating over a solid region in space.

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