Answer:
S = 2π/9 [³/₂√10 − ½ ln(-3 + √10)]
S ≈ 3.946
Step-by-step explanation:
For a curve rotated about the x-axis, the surface area is:
S = ∫ₐᵇ 2πy ds,
where ds = √(1 + (dy/dx)²) dx.
y = e⁻³ˣ
dy/dx = -3e⁻³ˣ
ds = √(1 + (-3e⁻³ˣ)²) dx
S = ∫₀°° 2πe⁻³ˣ √(1 + (-3e⁻³ˣ)²) dx
If u = -3e⁻³ˣ, then du = 9e⁻³ˣ dx, or du/9 = e⁻³ˣ dx.
When x = 0, u = -3. When x = ∞, u = 0.
S = ∫₋₃⁰ 2π √(1 + u²) (du/9)
S = 2π/9 ∫₋₃⁰ √(1 + u²) du
S = 2π/9 [ ½ u √(1 + u²) + ½ ln|u + √(1 + u²)| ] |₋₃⁰
S = 2π/9 {[0] − [ -³/₂√10 + ½ ln(-3 + √10) ]}
S = 2π/9 [³/₂√10 − ½ ln(-3 + √10)]
S ≈ 3.946
