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The problem you uploaded involves calculating integrals over a given region \( D \) in \( \mathbb{R}^2 \) and evaluating the integral of a function \( f(x, y) \). Here's the problem breakdown:

### Problem Statement:
You are asked to calculate the integral \(\iint_D f \, dA\) for the given region \( D \subset \mathbb{R}^2 \) and function \( f: D \rightarrow \mathbb{R} \).

#### Part (a):
- **Region \( D \)**: \( D = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 3^2\} \), which represents a disk centered at the origin with radius 3.
- **Function \( f(x, y) \)**: \( f(x, y) = \ln(1 + x^2 + y^2) \).

#### Part (b):
- **Region \( D \)**: \( D \) is the triangular region with vertices at \( (0,0) \), \( (2,0) \), and \( (0,1) \).
- **Function \( f(x, y) \)**: \( f(x, y) = e^{xy - y^2} \).

### Approach to Solving the Problem:

1. **Part (a):** Since the region is a disk, it is convenient to convert the double integral to polar coordinates.
   \[
   x = r\cos\theta, \quad y = r\sin\theta
   \]
   The integral becomes:
   \[
   \iint_D \ln(1 + r^2) \, r \, dr \, d\theta
   \]
   with \( r \) ranging from 0 to 3, and \( \theta \) ranging from 0 to \( 2\pi \).

2. **Part (b):** For the triangular region, the limits of integration depend on the vertices of the triangle. The integral can be set up using either \( dx \, dy \) or \( dy \, dx \) with the appropriate limits determined by the line equations of the triangle.

Would you like to see the detailed steps and final results for each part?

Calculate the double integral \( \iint_D f \, dA \) for the given regions \( D \subset \mathbb{R}^2 \) and functions \( f: D \rightarrow \mathbb{R} \).

This problem involves calculating double integrals over specified regions in \( \mathbb{R}^2 \), including a circular region and a triangular region. It covers the transformation to polar coordinates and applying the appropriate limits for integration.

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