I am trying to solve the following problem.
Consider a "cylinder" with base bounded by the polar curve $r(\theta)=1+\delta\sin(n\theta)$, where $n\in\mathbb{Z}$ and $\delta>0$ is sufficiently small. Find an approximation for the vertical portion of the cylinder (to error up to second order $\delta$).
I computed the arclength of the base which is needed for the surface area. which is $\int_0^{2\pi}\sqrt{(1+\frac{dr}{d\theta})}$ but the expression cannot be integrated nicely.