Final answer:
The triple integral over the cylindrical region can be evaluated using symmetry, resulting in the integral of the constant term 8 times the volume of the cylinder, which is 1536π.
Step-by-step explanation:
The student's question is about evaluating a triple integral over a cylindrical region using geometric interpretation and symmetry. The region is defined by x2 + y2 ≤ 16, −6 ≤ z ≤ 6. The integral is ∫ C (8 + 5x2yz2) dV.
To evaluate this using geometric interpretation, we note two things: the term 8 is constant throughout the volume of the cylinder, and the symmetry of the function 5x2yz2 about the z-axis means that its integral cancels out over the symmetric bounds of z. Since the cylinder is symmetric about the z-axis, for every point with a positive x, there is a point with a negative x at the same y and z, leading to a net zero contribution from the 5x2yz2 term.
Thus, the integral of 8 over the volume of the cylinder is straightforward: it is 8 times the volume of the cylinder. The volume of a cylinder is given by the area of its base (a circle with radius 4, since x2 + y2 ≤ 16) multiplied by its height (which is 12, from −6 ≤ z ≤ 6).
So, the integral evaluates to 8 × π × 42 × 12, which simplifies to 8 × π × 16 × 12, or 1536π.