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The problem involves evaluating a double integral of the function \( 4xy \) over the triangular region \( D \), with vertices at \( (0, 0) \), \( (1, 2) \), and \( (0, 3) \).

### Step 1: Identify the boundaries of the region \( D \)
We are given a triangular region with vertices at:
- \( (0, 0) \)
- \( (1, 2) \)
- \( (0, 3) \)

To describe this region in terms of inequalities, we first find the equation of the line between \( (0, 0) \) and \( (1, 2) \). The slope of this line is:
\[
m = \frac{2 - 0}{1 - 0} = 2
\]
Thus, the equation of the line is:
\[
y = 2x
\]

Now, the limits for \( x \) will range from \( 0 \) to \( 1 \) (the \( x \)-coordinates of the triangular region), and for a given \( x \), \( y \) will range between \( y = 2x \) and \( y = 3 \) (the vertical height of the triangle).

### Step 2: Set up the double integral
The double integral is set up as:
\[
\int_0^1 \int_{2x}^3 4xy \, dy \, dx
\]

### Step 3: Evaluate the inner integral (with respect to \( y \))
The inner integral is:
\[
\int_{2x}^3 4xy \, dy
\]
To solve this, treat \( x \) as a constant. The antiderivative of \( 4xy \) with respect to \( y \) is:
\[
\frac{4xy^2}{2} = 2xy^2
\]
Now, evaluate it at the bounds \( y = 3 \) and \( y = 2x \):
\[
\left[ 2x(3)^2 \right] - \left[ 2x(2x)^2 \right] = 18x - 16x^3
\]

### Step 4: Evaluate the outer integral (with respect to \( x \))
Now, we integrate \( 18x - 16x^3 \) with respect to \( x \):
\[
\int_0^1 (18x - 16x^3) \, dx
\]
The antiderivative of \( 18x \) is \( 9x^2 \), and the antiderivative of \( 16x^3 \) is \( 4x^4 \). So we have:
\[
9x^2 - 4x^4
\]
Evaluating this at the bounds \( x = 1 \) and \( x = 0 \):
\[
\left[ 9(1)^2 - 4(1)^4 \right] - \left[ 9(0)^2 - 4(0)^4 \right] = 9 - 4 = 5
\]

Thus, the value of the double integral is \( \boxed{5} \).

Would you like more details on any of these steps? Here are 5 related questions that expand on this topic:

1. How do we determine the boundaries of a triangular region for a double integral?
2. What are the general steps for evaluating a double integral over a region?
3. How do changes in the function being integrated affect the calculation of double integrals?
4. What is the significance of the order of integration (i.e., integrating with respect to \( y \) first, then \( x \))?
5. How does the shape of the region (e.g., triangular, rectangular) influence the setup of the limits for the integral?

**Tip:** Always visualize the region of integration before setting up the limits for the double integral. This can help avoid mistakes with the boundaries.

Evaluate the double integral of 4xy over the triangular region with vertices (0,0), (1,2), and (0,3).

This problem involves evaluating the double integral of the function 4xy over a triangular region with vertices at (0, 0), (1, 2), and (0, 3). The solution includes finding the bounds of integration and performing step-by-step integration.

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