Final answer:
To find the area of the surface obtained by rotating the curve y = 1^ex about the x-axis, we can use the formula for finding the surface area of a curve rotated about the x-axis.
Step-by-step explanation:
To find the exact area of the surface obtained by rotating the curve about the x-axis, we can use the formula for finding the surface area of a curve rotated about the x-axis: A = 2π∫aᵇ y√(1+(dy/dx)2) dx.
For the curve y = 1ex, we have y = f(x) = 1ex. Taking the derivative, we have dy/dx = f'(x) = 1ex. Plugging these values into the formula, we get:
A = 2π∫0³ 1ex√(1+(1ex)2) dx, where a = 0 and b = 3.
Therefore, the area of the lateral surface of each cylinder can be expressed as:
dA = 2πr(x) dx = 2π√(1 + ex) dx
The total surface area A is:
A = ∫₀⁵ 2π√(1 + ex) dx