Whoa, it's stronger than that. The *in*definite integrals are identical: In[183]:= Assuming[Abs@x <π,Simplify[ArcLength[Sin@t, {t, 0, x}] - ArcLength[Circle[{0, 0}, {1,√2}, {0, x}]]]] Out[183]= 0 This invites the animation of a pair of ellipses pinching the sine wave while bumpily rolling along it, with the point of contact and centers moving at a constant horizontal speed. I expect the ellipses to be in the "tall" phase at the extrema of the sinusoid. —rwg On Wed, Dec 16, 2020 at 2:22 AM Bill Gosper <billgosper@gmail.com> wrote:
The arclength of one period of sin x = the circumference of an ellipse with semiaxes 1 and √2.
In[112]:= #1 == #2 == FunctionExpand@#1== N@# &[ArcLength[Circle[{0, 0}, {√2, 1}]], ArcLength[Sin@x, {x, 0, 2 π]}]]
Out[112]= 4 EllipticE[-1] == 4 √2 EllipticE[1/2] == 4 √2 π^(3/2)/Gamma[1/4]^2 + Gamma[1/4]^2/√(2π) == 7.64039557805542
𝚪(¼) is the rightful value of the symbol 𝛕. And someone should write Beckmann II: A History of 𝛕. —rwg