To evaluate the integral

$\int_{\Omega} (x + y + z)^{3/2} \, dx \, dy \, dz,$

where $\Omega$ is defined by the constraints:

1. $x^2 + y^2 + z^2 \leq 12$ (a sphere of radius $\sqrt{12}$),
2. $x^2 + y^2 \leq 4z$ (a paraboloid),

we need to carefully analyze the region and use an appropriate coordinate system for simplification. A spherical or cylindrical coordinate system is likely useful here. Here's how to proceed:

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Step 1: Analyze the Region $\Omega$

• The sphere $x^2 + y^2 + z^2 = 12$ has a radius of $\sqrt{12} = 2\sqrt{3}$ and is centered at the origin.
• The paraboloid $x^2 + y^2 = 4z$ opens upwards and intersects the sphere to define the upper boundary.

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Step 2: Convert to Spherical Coordinates

In spherical coordinates:

• $x = \rho \sin\phi \cos\theta$,
• $y = \rho \sin\phi \sin\theta$,
• $z = \rho \cos\phi$,
• The volume element is $dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$.

Transform the boundaries:

1. $x^2 + y^2 + z^2 \leq 12$ becomes $\rho \leq 2\sqrt{3}$,
2. $x^2 + y^2 \leq 4z$ in spherical coordinates is $\rho^2 \sin^2\phi \leq 4\rho\cos\phi$, simplifying to $\rho \sin^2\phi \leq 4\cos\phi$.

This gives the effective constraint for $\phi$: $\sin^2\phi \leq \frac{4}{\rho}\cos\phi.$

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Step 3: Set Up the Integral in Spherical Coordinates

The integral becomes: $\int_0^{2\pi} \int_0^{\phi_{\max}} \int_0^{2\sqrt{3}} (\rho \sin\phi \cos\theta + \rho \sin\phi \sin\theta + \rho \cos\phi)^{3/2} \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta,$ where $\phi_{\max}$ depends on the intersection of the paraboloid and sphere.

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Step 4: Simplify and Compute (if feasible)

The angular dependence and the limits may require numerical evaluation. If you’re solving this manually, simplify the expression by:

• Splitting into symmetric components,
• Computing each term individually using symmetry,
• Substituting bounds explicitly.

Do you want me to compute this numerically or proceed analytically for detailed evaluation?

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Relative Questions

1. How do spherical coordinates simplify integrals over regions bounded by spheres or paraboloids?
2. Can the symmetry of the region reduce the complexity of this integral?
3. How do we determine the intersection of a sphere and a paraboloid?
4. What are typical strategies for evaluating integrals with $(x + y + z)^{3/2}$-type expressions?
5. How does the volume element $\rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$ come into play in spherical coordinates?

Tip: Always visualize the intersection of 3D regions to better understand