Perimeter of Lemniscate using Elliptic Integral of First Kind

This is the third video in a series of videos on elliptic integrals. The full playlist is available here:    • Elliptic Integrals   Previously we expressed the perimeter of the ellipse using elliptic integrals of the second kind. Now we will find the perimeter of the lemniscate using the elliptic integral of the first kind. Besides we will transform our integral into another form involving the integration of (1-y⁴)^(-1/2) from y=0 to y=1. Finally, by using the beta function, we will express the perimeter of the lemniscate in terms of π and Γ(1/4). In a later video, we will find the points of dimidiation and trisection of the portion of the lemniscate restricted to the first quadrant. TIMESTAMPS 0:00 Lemniscate definition 0:56 Cartesian equation and sketch 1:30 Using polar coordinates 6:00 Element of arclength 6:52 Perimeter of lemniscate 8:02 Rewriting integrand to have quartic under radical 11:06 Beta Function 12:46 Perimeter of lemniscate in terms of Γ- function