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To find the volume of the solid bounded by the paraboloid \( z = 4 - x^2 - y^2 \) and the \( xy \)-plane, we can set up a double integral in cylindrical coordinates.

### Solution:

1. **Convert to cylindrical coordinates:**
   The paraboloid is given by \( z = 4 - x^2 - y^2 \). In cylindrical coordinates, we have:
   \[
   x = r \cos \theta, \quad y = r \sin \theta, \quad \text{and} \quad z = 4 - r^2.
   \]

2. **Set up the bounds:**
   The solid is bounded by the \( xy \)-plane, which means \( z \geq 0 \). Setting \( z = 0 \) gives:
   \[
   4 - r^2 = 0 \Rightarrow r^2 = 4 \Rightarrow r = 2.
   \]
   Therefore, \( r \) ranges from \( 0 \) to \( 2 \), and \( \theta \) ranges from \( 0 \) to \( 2\pi \).

3. **Set up the integral:**
   The volume \( V \) is given by the double integral of \( z \) over the region \( R \) in the \( r \)-\( \theta \) plane:
   \[
   V = \int_0^{2\pi} \int_0^2 (4 - r^2) \, r \, dr \, d\theta.
   \]

4. **Evaluate the inner integral:**
   \[
   \int_0^2 (4 - r^2) \, r \, dr = \int_0^2 (4r - r^3) \, dr.
   \]
   Splitting this up, we have:
   \[
   \int_0^2 4r \, dr - \int_0^2 r^3 \, dr.
   \]
   Calculating each part:
   - For \( \int_0^2 4r \, dr \):
     \[
     \int 4r \, dr = 2r^2 \Big|_0^2 = 2 \times 4 = 8.
     \]
   - For \( \int_0^2 r^3 \, dr \):
     \[
     \int r^3 \, dr = \frac{r^4}{4} \Big|_0^2 = \frac{16}{4} = 4.
     \]
   So, the inner integral is:
   \[
   8 - 4 = 4.
   \]

5. **Evaluate the outer integral:**
   \[
   V = \int_0^{2\pi} 4 \, d\theta = 4 \theta \Big|_0^{2\pi} = 4 \times 2\pi = 8\pi.
   \]

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

Would you like more details on any step, or do you have any questions?

Here are five related questions for further exploration:

1. How would the setup change if the paraboloid equation were different, such as \( z = 6 - x^2 - y^2 \)?
2. How can cylindrical coordinates simplify the computation of volumes for other types of solids?
3. What if the paraboloid was bounded by a plane other than the \( xy \)-plane, like \( z = 1 \)?
4. How would we find the volume if the region had additional bounds within \( r \) or \( \theta \)?
5. How can we apply the same method to find the volume of a solid bounded by multiple surfaces?

**Tip:** When setting up integrals in cylindrical coordinates, always carefully determine the bounds for \( r \) and \( \theta \) to accurately represent the region in the \( xy \)-plane.

Find the volume of the solid bounded by the paraboloid z = 4 - x^2 - y^2 and the xy-plane.

Learn how to calculate the volume of the solid bounded by the paraboloid z = 4 - x^2 - y^2 and the xy-plane using cylindrical coordinates and double integrals. This solution walks through setting up and evaluating the integral to find the exact volume, covering key calculus concepts suitable for college students.

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