Verified answer

A = 2π ∫ {0 to ∞} e^(-6x) dx

Basically you're adding the circumference of many, many circles with infinitesimally small circumferences. Each circle's circumference is 2πr, where each one's radius is the distance from the curve to the axis of revolution.

A = 2π (lim b→∞ [ ∫ {0 to b} e^(-6x) dx)

For the integral, let u = -6x

du = -6 dx

-1/6 du = dx

A = 2π (lim b→∞ (-1/6) ∫ {0 to b} e^u du)

A = -π/3 (lim b→∞ [e^u]{0 to b}

A = -π/3 (lim b→∞ [e^(-6x)]{0 to b}

A = -π/3 (lim b→∞ [e^(-6b) - e^(0)]

A = -π/3 (lim b→∞ [1/e^(6b) - 1]

A = -π/3 (1/∞ - 1)

A = -π/3 (0 - 1)

A = π/3