Final answer:
The triple integral over the cylindrical region is simplified to calculating the volume of the cylinder multiplied by the constant 6 due to symmetry, with the x²yz² term canceling out.
Step-by-step explanation:
The triple integral ∫∫∫C(6+5x²yz²)dV, where C is the cylindrical region x²+y²≤16, -4≤z≤4, can be evaluated using geometric interpretation and symmetry. The integrand 6 is symmetric and its integral over the cylindrical volume is simply the constant times the volume of the cylinder. The term 5x²yz² will integrate to zero across the symmetric bounds for y and z because it will have symmetrically positive and negative contributions that cancel each other out. Hence, the integral simplifies to calculating the volume of a cylinder with radius 4 and height 8 (from z=-4 to z=4) and then multiplying by 6, which is 384π.