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The question asks to calculate the arc length of the curve \( y = x^2 - \frac{1}{8} \ln x \) from point \( A(1,1) \) to point \( B\left(3, 9 - \frac{1}{3} \ln 3\right) \).

### Formula for Arc Length:
The arc length \( L \) of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \) is computed as:

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

### Step 1: Find the derivative \( \frac{dy}{dx} \)
Given the function:
\[
y = x^2 - \frac{1}{8} \ln x
\]
Differentiate \( y \) with respect to \( x \):

\[
\frac{dy}{dx} = 2x - \frac{1}{8x}
\]

### Step 2: Set up the Arc Length Integral
The arc length formula becomes:

\[
L = \int_1^3 \sqrt{1 + \left( 2x - \frac{1}{8x} \right)^2} \, dx
\]

Now I will calculate this integral.The arc length of the curve \( y = x^2 - \frac{1}{8} \ln x \) from point \( A(1,1) \) to point \( B\left(3, 9 - \frac{1}{3} \ln 3\right) \) is:

\[
L = 8 + 0.125 \ln 3
\]

This is the exact arc length for the given curve over the specified interval.

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. How is the arc length formula derived for a given curve?
2. What happens if we approximate the arc length for a non-smooth function?
3. Can we calculate the arc length of a parametric curve using a similar formula?
4. How does the arc length change if the curve is modified by scaling in one direction?
5. Can the arc length formula be extended to surfaces in three dimensions?

**Tip:** Always check if the function is continuous and differentiable in the interval before applying the arc length formula!

Calculate the arc length of the curve y = x^2 - (1/8) ln x from point A(1,1) to point B(3, 9 - (1/3) ln 3).

This problem calculates the arc length of the curve y = x^2 - (1/8) ln x from A(1,1) to B(3, 9 - (1/3) ln 3) using the arc length formula and integral calculus. It involves differentiation and logarithmic functions, ideal for college-level students.

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