To find the length of the curve $x = \frac{y^3}{6} + \frac{1}{2y}$ for $1 \leq y \leq 2$, we use the formula for the arc length of a curve defined parametrically or as a function of $y$. The general formula for arc length when $x$ is a function of $y$ is:

$L = \int_{y_1}^{y_2} \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy$

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

Given: $x = \frac{y^3}{6} + \frac{1}{2y}$ Differentiate $x$ with respect to $y$:

$\frac{dx}{dy} = \frac{d}{dy} \left( \frac{y^3}{6} \right) + \frac{d}{dy} \left( \frac{1}{2y} \right)$ $\frac{dx}{dy} = \frac{3y^2}{6} - \frac{1}{2y^2} = \frac{y^2}{2} - \frac{1}{2y^2}$

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

Now square $\frac{dx}{dy}$:

$\left( \frac{dx}{dy} \right)^2 = \left( \frac{y^2}{2} - \frac{1}{2y^2} \right)^2$ This expands as:

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

Step 3: Set up the arc length integral

The arc length formula becomes:

$L = \int_1^2 \sqrt{1 + \left( \frac{y^4}{4} - \frac{1}{2} + \frac{1}{4y^4} \right)} \, dy$

Simplify the expression inside the square root:

$L = \int_1^2 \sqrt{\frac{y^4}{4} + \frac{1}{2} + \frac{1}{4y^4}} \, dy$

Step 4: Evaluate the integral

The integral does not have a simple elementary antiderivative, so it would generally be evaluated using numerical methods.

I can compute this integral numerically to give the final arc length.

Let me compute it.The length of the curve $x = \frac{y^3}{6} + \frac{1}{2y}$ for $1 \leq y \leq 2$ is approximately $1.417$ units.

Would you like more details on the process or have any other questions?

Here are some related questions for further exploration:

1. How does the arc length formula change if the curve is given as $y$ as a function of $x$?
2. Can you compute the arc length of a straight line using the same formula?
3. What is the arc length of the curve $x = y^2$ from $y = 0$ to $y = 1$?
4. How would the computation differ if $x = \frac{y^3}{6} + \frac{1}{2y}$ were parametrized differently?
5. Can the arc length formula be used to find surface areas of revolution?

Tip: For curves with complex derivatives, numerical methods such as Simpson's rule or Gaussian quadrature are commonly used to estimate arc lengths.