Let's proceed with simplifying and evaluating the surface area integral. The surface area $S$ is given by:

$S = 2\pi \int_{\frac{1}{2}}^{2} \left(\frac{x^3}{2} + \frac{1}{6x} - \frac{1}{2}\right) \sqrt{1 + \left(\frac{3x^2}{2} - \frac{1}{6x^2}\right)^2} \, dx$

Step 1: Simplify $\left(\frac{dy}{dx}\right)^2$

Let's first simplify the expression for $\left(\frac{dy}{dx}\right)^2$:

$\frac{dy}{dx} = \frac{3x^2}{2} - \frac{1}{6x^2}$

Squaring it:

$\left(\frac{dy}{dx}\right)^2 = \left(\frac{3x^2}{2} - \frac{1}{6x^2}\right)^2 = \frac{9x^4}{4} - \frac{1}{2} + \frac{1}{36x^4}$

So the term inside the square root is:

$1 + \left(\frac{dy}{dx}\right)^2 = 1 + \left(\frac{9x^4}{4} - \frac{1}{2} + \frac{1}{36x^4}\right) = \frac{9x^4}{4} + \frac{1}{2} + \frac{1}{36x^4}$

Step 2: Substitute into the Integral

The integral for surface area now becomes:

$S = 2\pi \int_{\frac{1}{2}}^{2} \left(\frac{x^3}{2} + \frac{1}{6x} - \frac{1}{2}\right) \sqrt{\frac{9x^4}{4} + \frac{1}{2} + \frac{1}{36x^4}} \, dx$

Step 3: Simplification Approach

This integral is complicated, so it might not simplify easily by elementary methods. However, to proceed symbolically:

• Notice that the integrand is difficult to simplify into a closed form.
• You could apply a substitution to simplify the integral, or alternatively, this could be approached using numerical integration techniques.

Given the complexity, do you want to proceed with a numerical approximation, or would you like me to explore another approach? This integral is typically solved using software for exact or approximate solutions.

Would you like an approximation or further exploration?

Related Questions:

1. What are the common techniques for solving integrals involving products of polynomials and radicals?
2. How does the surface area compare if the function were linear instead of cubic?
3. What methods can be used to simplify expressions involving $x^n$ and square roots?
4. How would this problem change if the limits were reversed?
5. How does changing the axis of revolution affect the surface area?

Tip:

When dealing with integrals that are challenging to solve analytically, consider using software like Wolfram Alpha, MATLAB, or other computational tools for exact or approximate evaluations.