Final answer:
To find the volume of the solid enclosed by the given surfaces, evaluate the triple integral with appropriate limits of integration.
Step-by-step explanation:
To find the volume of the solid enclosed by the given surfaces, we first need to determine the limits of integration. The region is bounded by the surfaces z = 6, y = 0, y = 16 - x², and z + y = 22. We can rewrite the equation y = 16 - x² as z = 22 - y to find the upper limit of z. Setting z = 6 and z + y = 22, we get y = 16. This means that the volume is bounded by y = 0, y = 16, z = 6, and z = 22 - y.
The volume can be computed using a triple integral, integrating with respect to x, y, and z. The limits of integration are:
- x: -4 to 4 (considering the symmetry of the solid)
- y: 0 to 16
- z: 6 to 22 - y
The integrand can be 1 (since we're just calculating the volume). Evaluating this triple integral will give us the volume of the solid enclosed by the surfaces.