To solve the problem of finding the surface area of the curve rotated about the y-axis, we use the formula for the surface area of a curve rotated around the y-axis:

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

The curve is defined parametrically as: $x = e^t - t, \quad y = 4e^{t/2}, \quad 0 \leq t \leq 1$

Steps:

1. Find derivatives $dx/dt$ and $dy/dt$:

   • For $x = e^t - t$: $\frac{dx}{dt} = e^t - 1$
   • For $y = 4e^{t/2}$: $\frac{dy}{dt} = 2e^{t/2}$

2. Compute the expression for $\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}$: $\left(\frac{dx}{dt}\right)^2 = (e^t - 1)^2$ $\left(\frac{dy}{dt}\right)^2 = (2e^{t/2})^2 = 4e^t$ $\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} = \sqrt{(e^t - 1)^2 + 4e^t}$

3. Set up the integral for surface area: Substituting $x = e^t - t$: $S = 2\pi \int_{0}^{1} (e^t - t) \sqrt{(e^t - 1)^2 + 4e^t} \, dt$

4. Solve the integral: This integral does not have a closed-form solution, so it would typically be evaluated numerically.

Let me know if you’d like me to compute the numerical value of the integral or explain the numerical process further.