To find the surface area of a surface generated by revolving a curve around an axis, we typically use the surface area formula from calculus. Here, I assume that the curve is being revolved around the x-axis (or another axis). The general formula for the surface area of a surface of revolution (around the x-axis) is:

$A = 2\pi \int_{a}^{b} y \cdot \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$

However, you have provided the equation $x = 4\sqrt{4 - y}$, which means $y$ is a function of $x$, so we need to rework this slightly.

Let’s first clarify some important details:

• We are given the equation $x = 4\sqrt{4 - y}$. We need to express $y$ as a function of $x$.
• We are also given the range of $y$: $0 \leq y \leq \frac{55}{16}$.

Step 1: Solve for $y$ in terms of $x$

We start with: $x = 4\sqrt{4 - y}$

Square both sides: $x^2 = 16(4 - y)$

Simplify: $x^2 = 64 - 16y$

Solve for $y$: $y = 4 - \frac{x^2}{16}$

Now we have $y(x)$.

Step 2: Determine the limits of integration

Since the range of $y$ is given as $0 \leq y \leq \frac{55}{16}$, we can plug these values into the equation $y = 4 - \frac{x^2}{16}$ to find the corresponding range for $x$.

For $y = 0$: $0 = 4 - \frac{x^2}{16} \implies x^2 = 64 \implies x = 8$

For $y = \frac{55}{16}$: $\frac{55}{16} = 4 - \frac{x^2}{16}$ Multiply through by 16: $55 = 64 - x^2 \implies x^2 = 9 \implies x = 3$

Thus, the limits of integration for $x$ will be from 3 to 8.

Step 3: Compute the derivative $\frac{dy}{dx}$

We know that: $y = 4 - \frac{x^2}{16}$

Differentiate $y(x)$ with respect to $x$: $\frac{dy}{dx} = -\frac{x}{8}$

Step 4: Apply the surface area formula

Now, using the formula for the surface area of revolution around the x-axis: $A = 2\pi \int_{3}^{8} y \cdot \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$

Substitute $y = 4 - \frac{x^2}{16}$ and $\frac{dy}{dx} = -\frac{x}{8}$: $A = 2\pi \int_{3}^{8} \left(4 - \frac{x^2}{16}\right) \cdot \sqrt{1 + \left(-\frac{x}{8}\right)^2} \, dx$

Simplify: $A = 2\pi \int_{3}^{8} \left(4 - \frac{x^2}{16}\right) \cdot \sqrt{1 + \frac{x^2}{64}} \, dx$

Step 5: Evaluate the integral

At this point, we can evaluate the integral, which may require numerical methods due to its complexity.

Would you like me to proceed with the numerical evaluation, or would you like further clarification on any of the steps so far?

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Here are five questions that expand on the problem:

1. How does changing the axis of revolution affect the surface area?
2. What happens if we change the function $x = 4\sqrt{4 - y}$ to a different power, such as $x = 4(4 - y)^n$?
3. Can we use parametric equations to simplify the surface area calculation?
4. What would the surface area be if we revolved the curve around the $y$-axis instead of the $x$-axis?
5. How does the range of $y$ affect the overall surface area?

Tip: For surface area problems, it's essential to correctly set up the limits of integration based on the problem's constraints.