To solve the problem of finding the average value of $f(x, y) = x^2 y$ over the rectangle $R$ with vertices at $(-4,0)$, $(-4,8)$, $(2,8)$, and $(2,0)$, we proceed as follows:

Formula for the Average Value of a Function:

The average value of a function $f(x, y)$ over a region $R$ is given by: $\text{Average Value} = \frac{1}{\text{Area of } R} \iint_R f(x, y) \, dA$

Step 1: Compute the Area of $R$

The vertices define a rectangle. The width is $2 - (-4) = 6$, and the height is $8 - 0 = 8$. Therefore, the area is: $\text{Area of } R = 6 \times 8 = 48$

Step 2: Set Up the Double Integral

The function $f(x, y) = x^2 y$ will be integrated over $R$: $\iint_R f(x, y) \, dA = \int_{x=-4}^{2} \int_{y=0}^{8} x^2 y \, dy \, dx$

Step 3: Solve the Inner Integral

First, integrate $x^2 y$ with respect to $y$: $\int_{y=0}^{8} x^2 y \, dy = x^2 \int_{y=0}^{8} y \, dy = x^2 \left[ \frac{y^2}{2} \right]_{0}^{8} = x^2 \left( \frac{8^2}{2} - \frac{0^2}{2} \right) = x^2 \cdot 32$

Thus, the inner integral evaluates to $32x^2$.

Step 4: Solve the Outer Integral

Now, integrate $32x^2$ with respect to $x$: $\int_{x=-4}^{2} 32x^2 \, dx = 32 \int_{x=-4}^{2} x^2 \, dx = 32 \left[ \frac{x^3}{3} \right]_{-4}^{2}$ $= 32 \left( \frac{2^3}{3} - \frac{(-4)^3}{3} \right) = 32 \left( \frac{8}{3} - \frac{-64}{3} \right) = 32 \cdot \frac{72}{3} = 32 \cdot 24 = 768$

Step 5: Compute the Average Value

Divide the integral by the area: $\text{Average Value} = \frac{\iint_R f(x, y) \, dA}{\text{Area of } R} = \frac{768}{48} = 16$

Final Answer:

The average value of $f(x, y)$ over $R$ is: $\boxed{16}$

Would you like a step-by-step explanation of any part?

Here are 5 related questions for further exploration:

1. How does the formula for the average value of a function change for non-rectangular regions?
2. Can the average value of a function over a region be negative? Why or why not?
3. How would the process differ if the function had a more complex form, such as $f(x, y) = \sin(xy)$?
4. What is the significance of the average value of a function in real-world applications?
5. How do you verify the bounds of integration for more complex regions?

Tip: Double-check the order of integration and bounds when evaluating multiple integrals.