To find the average value of the function $f(x, y) = e^y \sqrt{4x + e^y}$ over the rectangular region $R = [0, 3] \times [0, 2]$, we use the formula for the average value of a function over a region:

$\text{Average value} = \frac{1}{\text{Area of } R} \iint_R f(x, y) \, dA$

Step 1: Calculate the Area of $R$

The region $R = [0, 3] \times [0, 2]$ is a rectangle with dimensions 3 and 2, so its area is:

$\text{Area of } R = 3 \times 2 = 6$

Step 2: Set Up the Integral

The integral over $R$ can be expressed as:

$\iint_R f(x, y) \, dA = \int_0^2 \int_0^3 e^y \sqrt{4x + e^y} \, dx \, dy$

Step 3: Evaluate the Inner Integral with Respect to $x$

Let's first integrate $f(x, y) = e^y \sqrt{4x + e^y}$ with respect to $x$ from $x = 0$ to $x = 3$:

$\int_0^3 e^y \sqrt{4x + e^y} \, dx$

Using the substitution $u = 4x + e^y$, we get $du = 4 \, dx$, or $dx = \frac{du}{4}$. When $x = 0$, $u = e^y$, and when $x = 3$, $u = 4 \cdot 3 + e^y = 12 + e^y$. The inner integral becomes:

$\int_{e^y}^{12 + e^y} e^y \sqrt{u} \cdot \frac{1}{4} \, du = \frac{e^y}{4} \int_{e^y}^{12 + e^y} \sqrt{u} \, du$

Step 4: Evaluate the Integral $\int \sqrt{u} \, du$

The antiderivative of $\sqrt{u}$ is $\frac{2}{3} u^{3/2}$. Thus,

$\int_{e^y}^{12 + e^y} \sqrt{u} \, du = \left[ \frac{2}{3} u^{3/2} \right]_{e^y}^{12 + e^y} = \frac{2}{3} \left( (12 + e^y)^{3/2} - (e^y)^{3/2} \right)$

Step 5: Substitute Back and Complete the Integral

Substituting back, the inner integral becomes:

$\frac{e^y}{4} \cdot \frac{2}{3} \left( (12 + e^y)^{3/2} - (e^y)^{3/2} \right) = \frac{e^y}{6} \left( (12 + e^y)^{3/2} - (e^y)^{3/2} \right)$

Now, the average value of $f(x, y)$ over $R$ is:

$\frac{1}{6} \int_0^2 \frac{e^y}{6} \left( (12 + e^y)^{3/2} - (e^y)^{3/2} \right) \, dy$

This expression represents the average value of $f(x, y)$ over $R$. Calculating it by hand is challenging, so numerical integration would provide an exact answer.

Would you like further clarification on the steps or a numeric approximation?

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1. How would the calculation change if $R$ had different bounds?
2. What substitution techniques are useful for functions similar to $e^y \sqrt{4x + e^y}$?
3. How does changing the exponent in the square root impact integration?
4. What is the impact of the $e^y$ factor on the average value over larger regions?
5. What happens if we add constraints to $x$ or $y$ in region $R$?

Tip: Substitution is a useful tool for simplifying square root and exponential terms in integrals.