1. (6 points) Evaluate the triple integral
\[
\iiint_{E} x y z d V
\]
where $E$ is the region below $x+y+2 z=8$ in the first octant.

Step 1 ：First, express the limits of the triple integral in terms of the variables x, y, and z. Since the region E is in the first octant, all variables are non-negative. The equation of the plane can be rewritten as z = 4 - 0.5x - 0.5y. The limits of z are from 0 to 4 - 0.5x - 0.5y. The limits of y are from 0 to 8 - x and the limits of x are from 0 to 8.

Step 2 ：Next, express the triple integral as: \[ \int_{0}^{8} \int_{0}^{8-x} \int_{0}^{4-0.5x-0.5y} x y z dz dy dx \]

Step 3 ：Then, compute the triple integral. The function f is given by f = x*y*z. The integral with respect to z is given by z_int = 8*x*y*(-0.125*x - 0.125*y + 1)**2. The integral with respect to y is given by y_int = 0.03125*x*(8 - x)**4 + (8 - x)**3*(0.0833333333333333*x**2 - 0.666666666666667*x) + (8 - x)**2*(0.0625*x**3 - 1.0*x**2 + 4.0*x). The integral with respect to x is given by x_int = 91.0222222222205.

Step 4 ：Finally, the value of the triple integral is \(\boxed{91.0222222222205}\).