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The question asks to rewrite the given double integral in polar coordinates:

\[
\int_{-2}^{2} \int_{0}^{\sqrt{4-y^2}} 3\sqrt{x^2 + y^2} \, dx \, dy
\]

### Step 1: Convert to Polar Coordinates

- The integrand \( \sqrt{x^2 + y^2} \) becomes \( r \) in polar coordinates, because \( r = \sqrt{x^2 + y^2} \).
- The differential element \( dx \, dy \) converts to \( r \, dr \, d\theta \).
- The limits of integration for \( x \) and \( y \) define a region that is a semicircle with radius 2, centered at the origin.

### Step 2: Identify the Region of Integration

- The bounds for \( y \) are from -2 to 2, and for each value of \( y \), \( x \) ranges from 0 to \( \sqrt{4 - y^2} \), describing the right half of the circle \( x^2 + y^2 = 4 \) in Cartesian coordinates.
- In polar coordinates, this is the region where \( r \) goes from 0 to 2, and \( \theta \) goes from \( 0 \) to \( 2\pi \) for the entire circle, but since we only want the right half, the bounds for \( \theta \) are \( 0 \leq \theta \leq \pi \).

### Step 3: Rewriting the Integral

Thus, the integral becomes:

\[
\int_{0}^{\pi} \int_{0}^{2} 3r^2 \, dr \, d\theta
\]

### Step 4: Match with the Options

Looking at the options provided:

- **Option (d)** matches this form.

Therefore, the correct answer is **(d)**.

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Would you like further details or have any questions?

Here are some related questions to explore:

1. How do we convert rectangular coordinates to polar coordinates in general?
2. What does the Jacobian determinant represent in coordinate transformations?
3. Why does the integrand \( \sqrt{x^2 + y^2} \) simplify to \( r \) in polar coordinates?
4. Can you describe the limits of integration for a different region, such as a quarter-circle?
5. How would this problem change if the integrand were different, such as \( 1 \) or \( r^3 \)?

**Tip:** When converting to polar coordinates, always remember to multiply the integrand by the factor \( r \), which comes from the Jacobian of the transformation.

Rewrite using polar coordinates: ∫_{-2}^{2} ∫_{0}^{√(4 - y²)} 3√(x² + y²) dx dy

Learn how to convert a double integral in Cartesian coordinates into polar coordinates. This problem involves polar coordinate transformation, Jacobian determinant, and region identification. Suitable for Grades 11-12.

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