Question

If the infinite curve y = e−9x, x ≥ 0, is rotated about the x-axis, find the area of the resulting surface.

Answer 1

The area of the resulting surface is 0.

Step-by-step explanation:

To find the area of the surface formed by rotating the curve y = e^(-9x) about the x-axis, we can use the formula for the surface area of revolution. This formula is given by:

S = 2π∫[a,b] f(x)√(1+(f'(x))^2) dx

In this case, a = 0 because the curve is defined for x ≥ 0. To find b, we need to find the x-coordinate of the point where the curve intersects the x-axis, which is when y = 0. So we solve the equation e^(-9x) = 0, which gives x = 0. Plugging these values into the formula, we have:

S = 2π∫[0,0] e^(-9x)√(1+(-9e^(-9x))^2) dx

The integral evaluates to 0, so the area of the resulting surface is 0.

Answer 2

We are given with the equation
y = e^-9x with the restriction of x ≥ 0
If rotated about the x-axis the area of the resulting surface is
A = π ∫ y² dx
Substituing
A = π ∫ (e^-9x)² dx
A = π ∫ e^-18x dx
The limits are from 0 to positive infinity.