Putting the title to math, let's say there's a function $f(x)$ such that $\lim_{x \to 0^{+}} f(x) = \infty$. The limit can be to $0$ or $0^{-}$, but this is for simplicity. For the purpose of getting an infinitely small base, we can have base of width $2\theta - \theta = \theta$ and have $\theta$ approach $0$. The area is, then, $$A = \lim_{\theta \to 0^{+}} \int^{2\theta}_{\theta}f(x)dx$$ Are there any $f(x)$ besides ones which have any logarithms in their antiderivative so that $A$ is a nonzero constant? $f(x) = \frac1x$ and $f(x)=\cot x$ both have logarithms in their antiderivative, and $A = \ln2$ for both.
Point of the question is to find out functions where the "infinite height" doesn't dominate the "$0$ base" (which causes area to be infinite). Also functions that the "$0$ base" doesn't dominate the "infinite height" (which causes area to be $0$). I would be glad to clarify in the comments if anything is confusing.