Final answer:
To evaluate the given triple integral over the region E, we need to convert it into spherical coordinates and substitute the spherical expressions for x, y, and z. Then, we can evaluate the integral using the given limits of integration. The result will be the volume enclosed by the region.
Step-by-step explanation:
To evaluate the given integral, we need to calculate the triple integral over the region E described. The region is defined as being between the spheres rho = 4 and rho = 7 and above the cone φ = π/3.
We can convert the triple integral into spherical coordinates, where
x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), and z = ρcos(φ).
The limits of integration for ρ are from 4 to 7, for φ are from π/3 to π, and for θ are from 0 to 2π.
The integrand is xyz, so we substitute these spherical expressions for x, y, and z into the integral.
After evaluating the integral, the result will be the numerical value of the volume enclosed by the given region.