Question

Consider the challenge it would be to calculate the circumference of a circle without calculus (ignoring the "convenient" formulas... how did they get discovered?). Using a definite integral, how would you set up and prove that the circumference of a circle or radius r is C = 2πr? Now do it again using polar coordinates. Discuss the merits (pros and cons) of each method.

Answer 1

rope stretchers never seemd to mind doing it without calculus

Answer 2

an old guy named archimedes tried a limiting approach

Answer 3

You can cut that circle like this:

Now group all pieces.

Calculate a+b to get the circonference.