Sorry, i didn't know what delta x was called

Ok, here is my problem, in the following diagram the height (length) of the cylinder:

My calculations: $\displaystyle Rcos(\theta)-Rcos(\theta+\delta \theta)$= $\displaystyle Rcos(\theta)-R(cos(\theta)cos(\delta \theta)+sin(\theta)sin(\delta \theta)$

$\displaystyle =R(cos\theta-cos\theta+sin\theta*\delta\theta)$

$\displaystyle =Rsin\theta*\delta\theta$ (assumptions $\displaystyle cos\delta\theta $becomes 1, $\displaystyle sin\delta\theta$ becomes $\displaystyle \delta \theta $

But the book says that the height is R\delta \theta, what am I doing wrong here?