Many proofs for the area of a circle start with something like
$$ A(r) = \int_0^r 2 \pi t dt $$
such as at https://en.wikipedia.org/wiki/Area_of_a_disk#Onion_proof , but I don't understand how to get to this starting point.
Using the same notation at the wikipedia article, we can imagine the radial line made up of many infintesimally small fragments $h$ each at a distance $t$ from the center of the circle. Considering just one of these fragments, the area swept out by this fragment around the circle (i.e., the circumferential line length) is
$$ L = \int_0^{2\pi} ([t+h]-t) d \theta = \int_0^{2\pi} h d \theta = 2 \pi h$$
But obviously the circumference of a line should be $2 \pi t$.
I see that there are various ways to derive this, but I'm specifically trying to understand where my logic is wrong in applying this method of reasoning. I realize this is an elementary question, but I appreciate any kind help.
Thank you.
One begins reasoning this way: A circle (actually, you mean a disk) is radially symmetric, i.e., the same shape in every direction from its center. (A square or an oval is not radially symmetric.) Thus you reason that the only variable that changes is in the radial direction, not the angular direction. You then reason that the radially-symmetric shapes that can "cover" the disk are rings of different radii. To cover the entire disk you sum up an infinite number of these rings.
The area of the black ring is $2 \pi t\ dt$. You add those up, from $t = 0$ to $t = r$.
If this reasoning isn't clear, I urge you to speak to your teacher.