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The problem asks to find the surface area generated by rotating the given parametric curve about the y-axis. The equations are:

\[
x = e^t - t, \quad y = 4e^{t/2}, \quad 0 \leq t \leq 9
\]

### Step-by-step solution:

1. **Surface area of a curve rotated around the y-axis:**
   The formula for the surface area of a curve rotated around the y-axis is:

   \[
   A = 2\pi \int_{a}^{b} x(t) \cdot \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \, dt
   \]
   
   Here:
   - \( x(t) = e^t - t \)
   - \( y(t) = 4e^{t/2} \)
   - \( 0 \leq t \leq 9 \)

2. **Derivatives of \( x(t) \) and \( y(t) \):**
   We first need to compute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \):
   
   \[
   \frac{dx}{dt} = \frac{d}{dt}(e^t - t) = e^t - 1
   \]
   
   \[
   \frac{dy}{dt} = \frac{d}{dt}(4e^{t/2}) = 4 \cdot \frac{1}{2} e^{t/2} = 2e^{t/2}
   \]

3. **Substitute the derivatives:**
   Now substitute the expressions for \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) into the surface area formula:

   \[
   A = 2\pi \int_{0}^{9} \left( e^t - t \right) \cdot \sqrt{ \left( e^t - 1 \right)^2 + \left( 2e^{t/2} \right)^2 } \, dt
   \]
   
   Simplify the square root:

   \[
   A = 2\pi \int_{0}^{9} \left( e^t - t \right) \cdot \sqrt{ (e^t - 1)^2 + 4e^t } \, dt
   \]

4. **Next steps:**
   At this point, the integrand is complicated, and finding the exact integral may require numerical methods or a calculator.

   The expression to evaluate is:

   \[
   A = 2\pi \int_{0}^{9} \left( e^t - t \right) \cdot \sqrt{ (e^t - 1)^2 + 4e^t } \, dt
   \]

This integral would typically be computed using numerical techniques such as Simpson's rule or by using a tool like a calculator or software for numerical integration.

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Would you like to dive into any details, or do you have specific questions about this process?

Here are some related questions:
1. How does the formula for surface area change if the curve is rotated about the x-axis?
2. What is the significance of finding the derivative of the parametric equations?
3. How can we numerically evaluate this integral using a calculator or software?
4. Why is the square root term important in surface area calculations?
5. How would the result change if the bounds of integration were different?

**Tip**: In problems involving parametric equations, always remember to compute the derivatives accurately for correct application of integral formulas.

Find the surface area generated by rotating the given parametric curve about the y-axis. The equations are x = e^t - t, y = 4e^(t/2), 0 ≤ t ≤ 9.

Find the surface area generated by rotating the parametric curve defined by x = e^t - t and y = 4e^(t/2) over the interval 0 ≤ t ≤ 9 about the y-axis. This problem covers calculus concepts including parametric equations and surface area of revolution.

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