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

Step 1 ：This is a definite integral problem. The integral is from \(\frac{1}{2}\) to 1 of a function. The function is a product of a constant \(2\pi\) and a function which is a sum of two terms. The first term is \(\frac{x^{3}}{2}\) and the second term is \(\frac{1}{6}\). This function is multiplied by the square root of another function which is a sum of two terms. The first term is \(\frac{3x^{2}}{2}\) and the second term is \(-\frac{1}{6x^{2}}\). The square of this function is taken and then the square root of the result is taken.

Step 2 ：The definite integral of the given function from \(\frac{1}{2}\) to 1 is approximately \(1.602\).

Step 3 ：Final Answer: \(\boxed{1.602}\)