We are tasked with finding the area of the surface generated when the curve

$y = \frac{x^3}{3} + \frac{1}{4x}$

is revolved about the $x$-axis from $x = \frac{1}{2}$ to $x = 1$.

The formula for the surface area $A$ of a solid of revolution when a curve $y = f(x)$ is revolved about the $x$-axis is given by:

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

Step 1: Compute the derivative $\frac{dy}{dx}$

Given $y = \frac{x^3}{3} + \frac{1}{4x}$, we differentiate with respect to $x$:

$\frac{dy}{dx} = x^2 - \frac{1}{4x^2}$

Step 2: Set up the surface area integral

Now, we need to substitute $y$ and $\frac{dy}{dx}$ into the surface area formula:

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

This is a complex integral, and solving it analytically would involve expanding the expression inside the square root, simplifying, and possibly using numerical methods.

I can evaluate this integral to find the exact surface area. Let's proceed with that calculation.It seems that the integral is quite complex and requires numerical methods for an efficient solution. I will proceed by computing the surface area numerically to find the exact result. Let's do that.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you'd like, I can help guide you through the steps to perform this calculation using a numerical integration tool.

Would you like me to walk you through that? Feel free to ask questions!

Here are 5 related questions:

1. How is the surface area formula derived for a curve rotated about the x-axis?
2. What are the common methods for numerically solving integrals?
3. How does differentiation affect the complexity of surface area calculations?
4. What are some techniques for simplifying complex integrals?
5. Why do we use numerical methods for integrals that are difficult to solve analytically?

Tip: When using numerical methods, always double-check your integration limits and ensure you're using the correct formula for the desired geometric setup.