To find the volume of the solid enclosed by the paraboloid $x = y^2 + z^2$ and the plane $x = 16$, we can set up an integral in cylindrical coordinates. In cylindrical coordinates $(r, \theta, x)$, we have:

1. The paraboloid $x = y^2 + z^2$ can be rewritten as $x = r^2$.
2. The plane $x = 16$ gives us an upper bound for $x$.

Steps:

1. Set up the bounds:

   • For $x$, we have $r^2 \leq x \leq 16$.
   • For $r$, since $x = r^2 \leq 16$, we find $r \leq \sqrt{16} = 4$. So $0 \leq r \leq 4$.
   • For $\theta$, it goes around the entire circle, so $0 \leq \theta \leq 2\pi$.

2. Volume Integral: The volume $V$ of the solid is given by: $V = \int_{0}^{2\pi} \int_{0}^{4} \int_{r^2}^{16} r \, dx \, dr \, d\theta$ Here, the $r$ factor appears because of the cylindrical coordinate volume element $dV = r \, dx \, dr \, d\theta$.

3. Evaluate the Integral:

   • First, integrate with respect to $x$: $\int_{r^2}^{16} r \, dx = r \cdot (16 - r^2)$

   • Substitute back into the integral: $V = \int_{0}^{2\pi} \int_{0}^{4} r(16 - r^2) \, dr \, d\theta$

   • Separate the integrals: $V = \int_{0}^{2\pi} d\theta \int_{0}^{4} (16r - r^3) \, dr$

   • Integrate with respect to $r$: $\int_{0}^{4} (16r - r^3) \, dr = \left[ 8r^2 - \frac{r^4}{4} \right]_{0}^{4} = \left( 8 \cdot 16 - \frac{256}{4} \right) = 128 - 64 = 64$

   • Now, integrate with respect to $\theta$: $V = \int_{0}^{2\pi} 64 \, d\theta = 64 \cdot 2\pi = 128\pi$

Final Answer:

The volume of the solid enclosed by the paraboloid and the plane is: $V = 128\pi$

Would you like more details or have any questions on specific steps?

Here are some related questions:

1. How would the solution change if the paraboloid were rotated along a different axis?
2. What would the volume be if the plane were at $x = 25$ instead?
3. Can we solve this problem using Cartesian coordinates instead of cylindrical?
4. How would the problem change if the paraboloid had the equation $x = y^2 + z^2 + 1$?
5. What is the general formula for the volume of solids bounded by paraboloids and planes?

Tip: Always check whether switching to a different coordinate system (such as cylindrical) simplifies setting up integrals for symmetric regions.