Final answer:
To find the surface area of the solid of revolution generated by rotating the curve y = e^(-3x) about the x-axis, we use the surface area formula for solids of revolution and integrate by applying appropriate calculus techniques, resulting in the total surface area.
Step-by-step explanation:
The student is asking about the surface area of a solid of revolution, which is generated by rotating the curve y = e−³x, where x ≥ 0, about the x-axis. To calculate the area of the resulting surface, we need to use the formula for the surface area of a solid of revolution:
Surface Area (A) = 2π ∫ y ∙ sqrt(1 + (dy/dx)²) dx
First, we need to calculate the derivative dy/dx:
- dy/dx = d/dx [e−³x] = -3e−³x
Then, we incorporate this into the surface area formula:
A = 2π ∫ from 0 to ∞ e−³x ∙ sqrt(1 + (-3e−³x)²) dx
This integral can be evaluated using integration techniques suitable for a college-level calculus course. The result will give the total surface area of the solid of revolution.