To calculate the length of the curve given by the function $y = \frac{x^{3/2}}{2} + \frac{1}{6x}$ over the interval $3 \leq x \leq 5$, we use the formula for the length of a curve in Cartesian coordinates:

$L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx$

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

The function is $y = \frac{x^{3/2}}{2} + \frac{1}{6x}$. Differentiating with respect to $x$:

$\frac{dy}{dx} = \frac{3}{4}x^{1/2} - \frac{1}{6x^2}$

Step 2: Set up the integral

The formula for the length of the curve becomes:

$L = \int_3^5 \sqrt{1 + \left( \frac{3}{4}x^{1/2} - \frac{1}{6x^2} \right)^2} \, dx$

Step 3: Evaluate the integral

This integral can be evaluated numerically. Let me calculate it.