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.