Volume of revolution and surface of revolution

Volume of revolution is found easiest using the washer method here.

Volume = (pi) (integral of) y^2 dx, for x from 1 to infinity.
= (pi) (integral of) 1/x^2 dx
= (pi) [-1/x] from 1 to infinity
= pi

The surface area of revolution is found using:

surface area = (2pi) (integral of) y sqrt(1+dy/dx ^2) dx

But dy/dx = ln x, so sqrt(1 + (ln x)^2) is always equal or greater than 1.

The integrand, (1/x) sqrt (1 + (ln x)^2), is therefore always greater than 1/x

But we know that the integral of 1/x is ln x, which diverges when the upper limit is infinity.

It therefore follows that the integral (and therefore the surface area) is infinite.

Hope that helps.

Fibonacci