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.

  • $\begingroup$ You shouldn't be considering a small change of length $h$ unless you plan to sum up infinitely many of them, and I do not see any sums here $\endgroup$ – ASKASK Dec 17 '15 at 1:33
  • $\begingroup$ @ASKASK If h is infinitesimally small, isn't the "height" of the area effectively small enough that I have a line? Or is there a better approach? $\endgroup$ – Struggling snowman Dec 17 '15 at 1:33
  • $\begingroup$ What do you think you are measuring the "height" of? $\endgroup$ – ASKASK Dec 17 '15 at 1:34
  • $\begingroup$ @ASKASK The height h is the height of the infinitesimally small piece of the radial line. It seems to me that “infinitesimally small” can be the height of a circumferential line (i.e., essentially zero). But are you saying this cannot be the case? Is it possible to mathematically explain why not? $\endgroup$ – Struggling snowman Dec 17 '15 at 1:44
  • $\begingroup$ @Strugglingsnowman I think it would be helpful if you could add a sketch to your question, showing what you mean by "height of the infinitesimally small piece of the radial line", what $t$ is, etc. In particular it's not clear what the expression $[(t+h)-t]$ refers to geometrically. $\endgroup$ – mweiss Dec 17 '15 at 2:43

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 $t = r$. enter image description here

If this reasoning isn't clear, I urge you to speak to your teacher.

  • $\begingroup$ I think this misses the point of my question, since I state in the first sentence that I'm trying to understand how to get to such a starting point based on my aforementioned logic. Or maybe, since it already has 3 upvotes, I'm missing something about this answer? $\endgroup$ – Struggling snowman Dec 17 '15 at 2:20
  • $\begingroup$ How far back would you accept a "starting point"? After all, you were not clear on the difference between a circle and a disk. Do we have start way way back there? $\endgroup$ – David G. Stork Jan 7 '16 at 1:41

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