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To express the given triple integral in spherical coordinates, we need to analyze the integral limits and the function \( f(\rho, \theta, \phi) \).

### Step 1: Understanding the Limits
The region \( E \) is bounded by:
1. The hemisphere \( z = \sqrt{4 - x^2 - y^2} \), which is the top half of a sphere of radius 2 centered at the origin.
2. The \( xy \)-plane, which corresponds to \( z = 0 \).

In spherical coordinates, the volume element \( dV = \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi \).

### Step 2: Determine the Limits of Integration
1. **\( \theta \):** Since we are covering a full circular region in the \( xy \)-plane, \( \theta \) ranges from \( 0 \) to \( 2\pi \).
2. **\( \phi \):** This angle goes from the positive \( z \)-axis to the \( xy \)-plane. So, \( \phi \) ranges from \( 0 \) to \( \frac{\pi}{2} \).
3. **\( \rho \):** Since the sphere has radius 2, \( \rho \) ranges from \( 0 \) to \( 2 \).

Thus, we have:
- \( a = 2 \)
- \( b = \frac{\pi}{2} \)

### Step 3: Identify \( f(\rho, \theta, \phi) \)
In spherical coordinates, \( \sqrt{x^2 + y^2} = \rho \sin \phi \). This means \( f(\rho, \theta, \phi) \) should be proportional to \( \rho \sin \phi \) to match the form of the integral given.

Looking at the answer choices:
- The correct answer for \( f(\rho, \theta, \phi) \) is **(c) \( \rho \sin \phi \)**.

### Summary
- **\( a = 2 \)**
- **\( b = \frac{\pi}{2} \)**
- **\( f(\rho, \theta, \phi) = \rho \sin \phi \)** (Option C)

Would you like a further breakdown or have any questions?

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Here are 5 related questions to explore further:
1. How do we convert rectangular coordinates to spherical coordinates in general?
2. Why is the volume element in spherical coordinates \( \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi \)?
3. How would the limits change if the hemisphere had a different radius?
4. How can we integrate a function over a spherical region using spherical coordinates?
5. What if the region extended below the \( xy \)-plane—how would that affect \( \phi \)’s limits?

**Tip:** Always ensure your volume element \( dV \) matches the coordinate system you’re using to avoid mistakes in setting up integrals.

Let E be the solid region enclosed by the hemisphere z = √(4 - x^2 - y^2) and the xy-plane. Use spherical coordinates to express the triple integral. Find values for a, b, and f(ρ, θ, φ) from the options provided.

Solve a triple integral in spherical coordinates for the solid region enclosed by a hemisphere and the xy-plane. This problem involves finding integration limits for a, b, and f(ρ, θ, φ) using spherical coordinates. Detailed solution with core concepts in volume integration and coordinate transformations.

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