I understand that we're calculating the area of an infinitesimal polar rectangle, and summing up many of them. My teacher kind of glossed over why this produced rdrd$\theta$. I tried verifying by hand, and I got:
*theta is in radians $$ dA = \frac {dθ}{2\pi}(\pi(r+dr)^2 - \pi(r)^2)\\ dA = \frac {d\theta}{2}(2rdr + dr^2)\\ dA = rdrd\theta + \frac{dr^2d\theta}{2} $$
What happens to the second term? My gut tells me it has something to do with the fact that the latter term's multiplicity of differential variables (3) is higher than the former term's (2), so as you "zoom in", it becomes irrelevant. Kind of like how
$$ \lim_{x\to\infty} \frac{x^3+10000000x^2}{x^2} = x $$
because the second term in the numerator's growth just gets outpaced.
Could someone provide a stronger understanding of all this? As I think I've demonstrated, my grasp on it is a bit foggy.