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?