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To find the length of the arc for the given function:

\[
y = \frac{1}{8}(x^2 - 8 \ln(x))
\]

from \(x = 4\) to \(x = 5\), we can use the formula for the arc length of a function \( y = f(x) \):

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

### Step 1: Find \(\frac{dy}{dx}\)
We first need to compute the derivative of the function:

\[
y = \frac{1}{8}(x^2 - 8 \ln(x))
\]

The derivative is:

\[
\frac{dy}{dx} = \frac{1}{8} \left( \frac{d}{dx}(x^2) - \frac{d}{dx}(8 \ln(x)) \right)
\]

\[
\frac{dy}{dx} = \frac{1}{8} \left( 2x - \frac{8}{x} \right) = \frac{1}{8} \cdot \frac{2x^2 - 8}{x}
\]

\[
\frac{dy}{dx} = \frac{x}{4} - \frac{1}{x}
\]

### Step 2: Substitute \(\frac{dy}{dx}\) into the arc length formula
Now we substitute this derivative into the arc length formula:

\[
L = \int_4^5 \sqrt{1 + \left( \frac{x}{4} - \frac{1}{x} \right)^2} \, dx
\]

We will simplify this expression and calculate the integral.

### Step 3: Simplify the integrand
Let's first expand the square term:

\[
\left( \frac{x}{4} - \frac{1}{x} \right)^2 = \frac{x^2}{16} - \frac{2}{4} + \frac{1}{x^2}
\]

Thus, the integrand becomes:

\[
\sqrt{1 + \frac{x^2}{16} - \frac{2}{4} + \frac{1}{x^2}}
\]

### Step 4: Compute the definite integral
We now need to compute the integral:

\[
L = \int_4^5 \sqrt{1 + \frac{x^2}{16} - \frac{2}{x} + \frac{1}{x^2}} \, dx
\]

The result can be calculated using numerical methods as this integral is not easy to solve by hand.

Let me know if you'd like me to compute the final numerical result or if you'd like more details on the steps! 

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Here are some related questions to explore further:
1. How do you compute arc length for parametric functions?
2. What is the difference between calculating arc length in Cartesian and polar coordinates?
3. Can you explain how logarithmic differentiation works?
4. How does the second derivative test apply to concavity in arc length problems?
5. What is the significance of using definite integrals for arc length?

**Tip:** When dealing with arc length, check if a simpler reparametrization might reduce the complexity of the integral.

Find the length of the arc formed by y = (1/8)(x^2 - 8ln(x)) from x = 4 to x = 5.

Learn how to calculate the arc length of the function y = (1/8)(x^2 - 8ln(x)) from x = 4 to x = 5. This detailed solution uses calculus concepts, including differentiation and the arc length formula, and is suitable for high school and early college students.

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