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The problem asks us to find the volume of the solid bounded above by the plane \( z = 4 \) and below by the paraboloid \( z = 4 - x^2 - y^2 \) using a triple integral in cylindrical coordinates.

### Step 1: Express the equations in cylindrical coordinates
In cylindrical coordinates, we have:
\[
x = r \cos \theta, \quad y = r \sin \theta, \quad z = z, \quad \text{and} \quad x^2 + y^2 = r^2
\]
So the paraboloid equation \( z = 4 - x^2 - y^2 \) becomes:
\[
z = 4 - r^2
\]

### Step 2: Set up the limits for the integrals
- The \( z \)-limits will be from the paraboloid to the plane: \( 4 - r^2 \leq z \leq 4 \).
- The \( r \)-limits are determined by where the paraboloid intersects the plane \( z = 0 \). Setting \( 4 - r^2 = 0 \) gives \( r = 2 \), so \( 0 \leq r \leq 2 \).
- The \( \theta \)-limits are \( 0 \leq \theta \leq 2\pi \) since the region is symmetric around the \( z \)-axis.

### Step 3: Write the volume integral
The volume \( V \) can be found by the triple integral:
\[
V = \int_0^{2\pi} \int_0^2 \int_{4 - r^2}^4 r \, dz \, dr \, d\theta
\]
Here, the \( r \) in the integrand accounts for the Jacobian when converting to cylindrical coordinates.

### Step 4: Evaluate the integral
Let's evaluate the integral step by step:

1. **Integrate with respect to \( z \):**
   \[
   \int_{4 - r^2}^{4} dz = [z]_{4 - r^2}^{4} = 4 - (4 - r^2) = r^2
   \]

   So the integral simplifies to:
   \[
   V = \int_0^{2\pi} \int_0^2 r^3 \, dr \, d\theta
   \]

2. **Integrate with respect to \( r \):**
   \[
   \int_0^2 r^3 \, dr = \left[\frac{r^4}{4}\right]_0^2 = \frac{16}{4} = 4
   \]

   Thus, the volume integral becomes:
   \[
   V = \int_0^{2\pi} 4 \, d\theta
   \]

3. **Integrate with respect to \( \theta \):**
   \[
   \int_0^{2\pi} 4 \, d\theta = 4\theta \bigg|_0^{2\pi} = 8\pi
   \]

### Final Answer
The volume of the solid is \( V = 8\pi \) cubic units.

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

### Relative Questions
1. How would you convert a different surface equation to cylindrical coordinates?
2. What would be the effect on the volume if the plane were at a different height, say \( z = 6 \)?
3. How do you determine the limits of integration in cylindrical coordinates for more complex regions?
4. What is the significance of the Jacobian in the conversion to cylindrical coordinates?
5. How would the problem change if the region was rotated or shifted in space?

### Tip
Always check that your limits of integration match the geometry of the problem, especially when dealing with bounds in different coordinate systems.

Use a triple integral in cylindrical coordinates to find the volume of the solid bounded above by the plane z = 4 and bounded below by the paraboloid z = 4 - x^2 - y^2.

Find the volume of the solid bounded by the plane z = 4 and the paraboloid z = 4 - x^2 - y^2 using triple integrals in cylindrical coordinates. A detailed solution involves setting up limits of integration and performing the integral step by step.

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