An integral theorem of Gauss

Gauss's 1818 theorem shows an integral remains unchanged when variables are replaced with their arithmetic and geometric means. By repeatedly applying this process to the example integral, the values of x=3 and y=7 eventually converge to 4.789013, called the arithmetic-geometric mean (AGM). This mathematical property provides an efficient way to evaluate elliptic integrals, as the original integral equals π/(2×AGM(x,y)).