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.