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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?

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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.

The solid enclosed by the paraboloid x = y^2 + z^2 and the plane x = 16

Learn how to find the volume of a solid bounded by the paraboloid x = y^2 + z^2 and the plane x = 16. This problem uses cylindrical coordinates to set up and evaluate the integral, suitable for college-level calculus students.

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