Step 1: Problem Definition:
Determine the surface area of a solid generated by revolving a parametrically defined curve around the y-axis.
Step 2: Mathematical Formulation:
The surface area \(S\) of a parametric curve \(x(t), y(t)\) revolved around the y-axis from \(t=a\) to \(t=b\) is given by:\[ S = \int_a^b 2\pi x(t) \, ds \]where \(ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\) represents the arc length element.
Step 3: Calculation Process:
1. Compute Derivatives:
\( \frac{dx}{dt} = e^t(\cos t - \sin t) \)
\( \frac{dy}{dt} = e^t(\sin t + \cos t) \)
2. Calculate Arc Length Element \(ds\):
\( \left(\frac{dx}{dt}\right)^2 = e^{2t}(1 - \sin(2t)) \)
\( \left(\frac{dy}{dt}\right)^2 = e^{2t}(1 + \sin(2t)) \)
\( \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = 2e^{2t} \)
\( ds = \sqrt{2} e^t \, dt \)
3. Integrate for Surface Area:
The integral is: \( S = \int_{0}^{\pi/2} 2\pi (e^t \cos t) (\sqrt{2} e^t \, dt) = 2\sqrt{2}\pi \int_{0}^{\pi/2} e^{2t} \cos t \, dt \).
Using the integration formula \( \int e^{at}\cos(bt) dt = \frac{e^{at}}{a^2+b^2}(a\cos(bt) + b\sin(bt)) \) with \(a=2\) and \(b=1\):
\( \int e^{2t} \cos t \, dt = \frac{e^{2t}}{5}(2\cos t + \sin t) \).
Evaluating the definite integral:
\( \left[ \frac{e^{2t}}{5}(2\cos t + \sin t) \right]_{0}^{\pi/2} = \frac{e^{\pi}}{5} - \frac{2}{5} = \frac{e^\pi - 2}{5} \).
4. Compute Final Surface Area:
\( S = 2\sqrt{2}\pi \left( \frac{e^\pi - 2}{5} \right) = \frac{2\sqrt{2}\pi}{5}(e^\pi - 2) \).
Step 4: Conclusion:
The calculated surface area is \( \frac{2\sqrt{2}}{5}\pi(e^\pi - 2) \) square units.