We are asked to evaluate the following triple integral in cylindrical coordinates:

$\int_{0}^{3} \int_{0}^{\sqrt{2}} \int_{x}^{\sqrt{16 - x^2}} e^{-x^2 - y^2} \, dy \, dx \, dz$

Step 1: Express the integral in cylindrical coordinates

In cylindrical coordinates, we have the following relationships:

• $x = r \cos \theta$
• $y = r \sin \theta$
• $z = z$ (unchanged)
• The area element $dx\,dy$ becomes $r \, dr \, d\theta$ due to the Jacobian in cylindrical coordinates.

Moreover, $x^2 + y^2 = r^2$, and thus the exponential term $e^{-x^2 - y^2}$ transforms to $e^{-r^2}$.

Step 2: Boundaries for cylindrical coordinates

We need to convert the limits of integration:

• $z$ runs from 0 to 3.
• The $xy$-region is circular, as $x^2 + y^2 \leq 16$, so $r$ ranges from 0 to 4 (since $\sqrt{16} = 4$).
• The angle $\theta$ will range from 0 to $2\pi$ for a full circular region.

Thus, the integral becomes:

$\int_0^3 \int_0^{4} \int_0^{2\pi} r e^{-r^2} \, d\theta \, dr \, dz$

Step 3: Evaluate the integral over $\theta$

The $\theta$-integral is straightforward:

$\int_0^{2\pi} d\theta = 2\pi$

This simplifies the integral to:

$2\pi \int_0^3 \int_0^4 r e^{-r^2} \, dr \, dz$

Step 4: Evaluate the radial integral

We now focus on the radial part:

$\int_0^4 r e^{-r^2} \, dr$

This integral can be solved using substitution. Let $u = r^2$, so $du = 2r \, dr$, or $\frac{du}{2} = r \, dr$. The limits for $u$ will go from 0 to $16$ as $r$ ranges from 0 to 4.

Thus, the radial integral becomes:

$\frac{1}{2} \int_0^{16} e^{-u} \, du$

The integral of $e^{-u}$ is simply $-e^{-u}$, so:

$\frac{1}{2} \left[ -e^{-u} \right]_0^{16} = \frac{1}{2} \left( -e^{-16} + e^0 \right) = \frac{1}{2} \left( 1 - e^{-16} \right)$

Step 5: Putting it all together

Now, the remaining integral is:

$2\pi \int_0^3 \frac{1}{2} \left( 1 - e^{-16} \right) \, dz$

This simplifies to:

$\pi \left( 1 - e^{-16} \right) \int_0^3 dz = \pi \left( 1 - e^{-16} \right) \cdot 3$

Thus, the final answer is:

$\boxed{3\pi \left( 1 - e^{-16} \right)}$

Would you like more details on any part of this solution? Here are some related questions to practice:

1. How do you compute integrals involving cylindrical coordinates?
2. What are common transformations for triple integrals into cylindrical coordinates?
3. Why does the Jacobian in cylindrical coordinates introduce an extra $r$?
4. How do you handle substitution in exponential integrals like $\int r e^{-r^2} \, dr$?
5. What is the significance of boundary conditions when converting to cylindrical coordinates?

Tip: Always check the region of integration carefully when switching to cylindrical or spherical coordinates—it can greatly simplify the integral!