To convert the given integral to spherical coordinates, we follow these steps:

Step 1: Spherical Coordinates Relations

The spherical coordinates $(\rho, \theta, \phi)$ are related to Cartesian coordinates $(x, y, z)$ as: $x = \rho \sin\phi \cos\theta, \quad y = \rho \sin\phi \sin\theta, \quad z = \rho \cos\phi,$ with the corresponding volume element: $dx \, dy \, dz = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta.$

Step 2: Identifying the Boundaries

The given Cartesian integral is: $\int_0^1 \int_0^{\sqrt{1-x^2}} \int_0^{\sqrt{4-x^2-y^2}} \frac{x^2 y}{\sqrt{x^2 + y^2}} \, dz \, dy \, dx.$

• Outer integral ($x$): $x$ ranges from $0$ to $1$.
• Middle integral ($y$): For a fixed $x$, $y$ ranges from $0$ to $\sqrt{1-x^2}$.
• Inner integral ($z$): For fixed $x$ and $y$, $z$ ranges from $0$ to $\sqrt{4-x^2-y^2}$.

These limits represent a region in the first octant of a sphere of radius $2$.

Step 3: Converting the Limits to Spherical Coordinates

• $\rho$: The radial distance ranges from $0$ to $2$, as the sphere has radius $2$.
• $\theta$: The azimuthal angle ranges from $0$ to $\frac{\pi}{2}$, since the region is in the first octant.
• $\phi$: The polar angle ranges from $0$ to $\frac{\pi}{2}$, again due to the first-octant restriction.

Step 4: Converting the Integrand

The integrand $\frac{x^2 y}{\sqrt{x^2 + y^2}}$ becomes: $x^2 y = (\rho \sin\phi \cos\theta)^2 (\rho \sin\phi \sin\theta),$ and $\sqrt{x^2 + y^2} = \rho \sin\phi$. Substituting, the integrand in spherical coordinates is: $\frac{x^2 y}{\sqrt{x^2 + y^2}} = \frac{(\rho \sin\phi \cos\theta)^2 (\rho \sin\phi \sin\theta)}{\rho \sin\phi} = \rho^2 \sin^2\phi \cos^2\theta \sin\theta.$

Step 5: Volume Element

The volume element $dx \, dy \, dz$ becomes $\rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$.

Step 6: The Final Integral

Substituting everything into spherical coordinates: $\int_0^1 \int_0^{\sqrt{1-x^2}} \int_0^{\sqrt{4-x^2-y^2}} \frac{x^2 y}{\sqrt{x^2 + y^2}} \, dz \, dy \, dx$ becomes: $\int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \int_0^2 \rho^4 \sin^3\phi \cos^2\theta \sin\theta \, d\rho \, d\phi \, d\theta.$

Let me know if you'd like additional details or have further questions!

Follow-Up Questions:

1. How are spherical coordinates derived from Cartesian coordinates?
2. What are the general integration limits for spherical coordinates?
3. Why does the volume element include $\rho^2 \sin\phi$?
4. How does the first-octant restriction influence the limits?
5. How would this integral change if the region was in the full sphere?

Tip:

Always carefully analyze the geometry of the region when converting integrals to spherical coordinates.