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To find the surface area generated by rotating the given parametric curve about the \( y \)-axis, we use the formula for the surface area of a parametric curve rotated about the \( y \)-axis:

\[
S = 2\pi \int_{a}^{b} x(t) \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt
\]

where \( x = e^t - t \), \( y = 4e^{t/2} \), and \( t \) ranges from \( 0 \) to \( 8 \).

### Step 1: Calculate \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)
- \( x = e^t - t \), so:
  \[
  \frac{dx}{dt} = e^t - 1
  \]

- \( y = 4e^{t/2} \), so:
  \[
  \frac{dy}{dt} = 4 \cdot \frac{1}{2}e^{t/2} = 2e^{t/2}
  \]

### Step 2: Compute the expression under the square root
The integrand's expression under the square root is:
\[
\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 = (e^t - 1)^2 + (2e^{t/2})^2
\]

### Step 3: Set up the integral for the surface area
The surface area integral becomes:
\[
S = 2\pi \int_{0}^{8} (e^t - t) \sqrt{(e^t - 1)^2 + (2e^{t/2})^2} \, dt
\]

This integral can be evaluated either numerically or using a computer algebra system, as it is not straightforward to solve analytically.

Would you like a numerical evaluation of this integral, or do you want more details on solving it? 

### Related Questions
1. What is the formula for the surface area when rotating about the \( x \)-axis?
2. How would the problem change if the rotation was about the \( x \)-axis instead of the \( y \)-axis?
3. Can this integral be simplified using trigonometric substitutions?
4. What is the geometric interpretation of rotating a parametric curve?
5. How would the problem differ if we rotated around a line other than the coordinate axes?

### Tip
For complex integrals, consider numerical methods or software tools like MATLAB, Mathematica, or Python to obtain approximate solutions.

Find the surface area generated by rotating the curve x = e^t - t, y = 4e^(t/2), 0 ≤ t ≤ 8 about the y-axis.

Learn how to compute the surface area generated by rotating the parametric curve x = e^t - t, y = 4e^(t/2) for 0 ≤ t ≤ 8 about the y-axis. This involves parametric differentiation and integration. Suitable for undergraduate calculus students.

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