Final answer:
To evaluate this triple integral, determine the limits of integration for each variable (x, y, and z) based on the given region. Then set up the integral using these limits and the given integrand (z). Simplify the integral to find a numerical value.
Step-by-step explanation:
To evaluate the given integral, we need to find the limits of integration for each variable: x, y, and z. Since the integral is to be evaluated in the first octant, the limits for x and y can be determined by the equations of the spheres: x² + y² + z² = 4 and x² + y² + z² = 16. In the first octant, x ranges from 0 to the square root of 4, which is 2, and y ranges from 0 to the square root of (4 - x²). As for z, it can range from 0 to the square root of (4 - x² - y²). The integral can now be set up as: ∫∫∫ₑ z dV, where dV = dx dy dz.
∫∫∫ₑ z dV = ∫₀² ∫₀√(4-x²) ∫₀√(4-x²-y²) z dz dy dx.
Simplifying this triple integral would involve evaluating it using the given limits and the function z, resulting in a numerical value.