Find the surface area

Question

Find the surface area generated by rotating the given curve about the y-axis.

x=e^(t)-t y=4e^(t/2) 0≤ t ≤ 8

Answer 1

Surface area A = 2π∫₀⁸x(t)√(x'²(t) + y'²(t))dt = 2π∫₀⁸[(e^t - t)√(e^t - 1)² + (2e^t/2)²]dt = 2π _∫0⁸(e^t - t)(e^t + 1)dt =

2π∫₀⁸(e^2t - te^t + e^t - t)dt = 2π(1/2e^2t + 2e^t - 1/2t² - te^t)₀⁸ = 2π[(1/2e¹⁶ + 2e⁸ -1/2·64 - 8e⁸) - (1/2 + 2)] = 8850318.025π = 27803943.29