To solve for the length of the curve, we need to follow specific steps. First, let's start with the equation you provided:

$8x = 4e^{2y} + e^{-2y}.$

Step 1: Solve for $x$

We start by isolating $x$: $x = \frac{1}{8} \left( 4e^{2y} + e^{-2y} \right).$

We can rewrite this as: $x = \frac{1}{2} e^{2y} + \frac{1}{8} e^{-2y}.$

Step 2: Differentiate $x$ with respect to $y$

Next, we'll differentiate $x$ with respect to $y$ to find $\frac{dx}{dy}$. Using the chain rule: $\frac{dx}{dy} = \frac{d}{dy} \left( \frac{1}{2} e^{2y} + \frac{1}{8} e^{-2y} \right).$

Differentiating each term: $\frac{dx}{dy} = \frac{1}{2} \cdot 2e^{2y} - \frac{1}{8} \cdot 2e^{-2y},$ $\frac{dx}{dy} = e^{2y} - \frac{1}{4} e^{-2y}.$

Step 3: Use the arc length formula

The formula for the length of a curve parameterized by $y$ is given by: $L = \int_{y_1}^{y_2} \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy.$

Substitute $\frac{dx}{dy}$ into the equation: $L = \int_{\ln 2}^{\ln 3} \sqrt{1 + \left( e^{2y} - \frac{1}{4} e^{-2y} \right)^2} \, dy.$

Step 4: Simplify the expression inside the square root

Now expand and simplify the expression: $\left( e^{2y} - \frac{1}{4} e^{-2y} \right)^2 = e^{4y} - 2 \cdot \frac{1}{4} \cdot e^{2y} \cdot e^{-2y} + \frac{1}{16} e^{-4y},$ $= e^{4y} - \frac{1}{2} + \frac{1}{16} e^{-4y}.$

Substitute this back into the arc length formula: $L = \int_{\ln 2}^{\ln 3} \sqrt{1 + e^{4y} - \frac{1}{2} + \frac{1}{16} e^{-4y}} \, dy,$ $L = \int_{\ln 2}^{\ln 3} \sqrt{\frac{1}{2} + e^{4y} + \frac{1}{16} e^{-4y}} \, dy.$

Step 5: Compute the integral

The next step would be to compute this integral, which might require numerical methods due to the complexity of the integrand. However, the setup is now complete, and you can evaluate this integral using a calculator or numerical tool.

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Would you like to go through this in more detail, or should we focus on calculating the integral numerically? Let me know if you have questions.

Follow-up Questions:

1. How would you solve for $y$ from the original equation?
2. What are some methods to simplify integrals with exponential terms?
3. Can you describe the general steps in finding the length of a curve?
4. How does the chain rule apply in differentiating parametric equations?
5. What numerical methods can be used to evaluate complex integrals?

Tip:

When dealing with parametric curves, always simplify the derivative as much as possible before applying the arc length formula to reduce the complexity of the integral.