To approximate the double integral of the function $f(x, y) = x(1 + y)$ over the region $D$, we can use a Riemann sum. The region $D$ is defined by the rectangle determined by the inequalities $-1 \leq x \leq 1$ and $2 \leq y \leq 4$.

Step-by-step procedure for the Riemann sum approximation:

1. Subdivide the region $D$:

   • $D$ is the rectangle defined by $-1 \leq x \leq 1$ and $2 \leq y \leq 4$, with side lengths of 2 units in the $x$-direction and 2 units in the $y$-direction.
   • We will divide the rectangle into squares with side length 1. This results in 4 squares, since the rectangle has dimensions $2 \times 2$.

   Specifically, the squares have the following coordinates (with their corners listed as $(x, y)$):

   • Square 1: $(-1, 2)$ to $(0, 3)$
   • Square 2: $(0, 2)$ to $(1, 3)$
   • Square 3: $(-1, 3)$ to $(0, 4)$
   • Square 4: $(0, 3)$ to $(1, 4)$

2. Determine the sample points:

   • The Riemann sum uses the top-left corner of each square as the sample point.
   • The top-left corners of the squares are:
     1. Square 1: $(-1, 3)$
     2. Square 2: $(0, 3)$
     3. Square 3: $(-1, 4)$
     4. Square 4: $(0, 4)$

3. Evaluate $f(x, y) = x(1 + y)$ at each sample point:

   • For $(-1, 3)$: $f(-1, 3) = -1(1 + 3) = -1(4) = -4$
   • For $(0, 3)$: $f(0, 3) = 0(1 + 3) = 0$
   • For $(-1, 4)$: $f(-1, 4) = -1(1 + 4) = -1(5) = -5$
   • For $(0, 4)$: $f(0, 4) = 0(1 + 4) = 0$

4. Compute the Riemann sum:

   • Since each square has an area of 1 (side length $1 \times 1$), the Riemann sum approximation of the double integral is: $R = (-4 \times 1) + (0 \times 1) + (-5 \times 1) + (0 \times 1) = -4 + 0 - 5 + 0 = -9$

Thus, the Riemann sum approximation of the double integral is $\boxed{-9}$.

Would you like more details on any of these steps?

Here are 5 questions that could further explore this concept:

1. How would the result change if we used the midpoint of each square instead of the top-left corner?
2. What happens if we increase the number of subdivisions in the region $D$?
3. How does this approximation compare with the exact value of the double integral?
4. What is the significance of the choice of sample points in a Riemann sum?
5. Can we approximate the integral using other methods, such as the trapezoidal rule?

Tip: Increasing the number of subdivisions in the region tends to improve the accuracy of the Riemann sum approximation for integrals.