To find the average value of a function $f(x, y) = e^y \sqrt{6x + e^y}$ over the rectangle $R = [0, 2] \times [0, 4]$, we use the formula:

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

Step 1: Calculate the Area of $R$

The rectangle $R$ has dimensions $[0, 2]$ in the $x$-direction and $[0, 4]$ in the $y$-direction. Thus, the area of $R$ is: $\text{Area of } R = (2 - 0)(4 - 0) = 8$

Step 2: Set Up the Double Integral

The average value of $f(x, y)$ over $R$ is: $\frac{1}{8} \iint_R e^y \sqrt{6x + e^y} \, dA$

In rectangular coordinates, this becomes: $\frac{1}{8} \int_0^4 \int_0^2 e^y \sqrt{6x + e^y} \, dx \, dy$

Step 3: Integrate with Respect to $x$

To simplify the integral, we first integrate with respect to $x$: $\int_0^2 e^y \sqrt{6x + e^y} \, dx$

Let $u = 6x + e^y$, so $du = 6 \, dx$ or $dx = \frac{du}{6}$.

When $x = 0$, $u = e^y$. When $x = 2$, $u = 12 + e^y$.

Thus, the integral with respect to $x$ becomes: $\int_{e^y}^{12 + e^y} e^y \sqrt{u} \cdot \frac{1}{6} \, du = \frac{e^y}{6} \int_{e^y}^{12 + e^y} \sqrt{u} \, du$

Step 4: Solve the Integral

Now, evaluate $\int \sqrt{u} \, du$: $\int \sqrt{u} \, du = \frac{2}{3} u^{3/2}$

Applying this to the limits $u = e^y$ to $u = 12 + e^y$: [ \frac{e^y}{6} \cdot \frac{2}{3} \left[(12 + e^y)^{3/2} - (e^y)^{3/2}\right] = \frac{e^y}{9} \left[(12 + e^y)^{3/2} - (e^y)^{3/2}\right] ]

Step 5: Integrate with Respect to $y$

Now we compute: $\frac{1}{8} \int_0^4 \frac{e^y}{9} \left[(12 + e^y)^{3/2} - (e^y)^{3/2}\right] dy$

Simplifying, the average value of $f(x, y)$ is: $\frac{1}{72} \int_0^4 e^y \left[(12 + e^y)^{3/2} - (e^y)^{3/2}\right] \, dy$

At this stage, the integral can be evaluated through further substitution or numerical methods.

Would you like a numerical approximation for the result or additional help with simplification?

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1. How is the average value of a function over a region calculated?
2. What are the steps to set up a double integral over a rectangle?
3. How does substitution simplify integrals?
4. What is the role of rectangular coordinates in double integrals?
5. How can numerical integration be applied to approximate complex integrals?

Tip: For complex integrals, consider numerical methods like Simpson's rule for approximation if analytic solutions are challenging.