Area of a circle "ds" over the whole circle formula_33 is just the arc length, which is its circumference, so this shows that the area "A" enclosed by the circle is equal to formula_34 times the circumference of the circle. Note that the area of a semicircle of radius r can be computed by the integral formula_35. By trigonometric substitution, we substitute formula_36, hence formula_37 The last step follows since the trigonometric identity formula_42 implies that formula_43 and formula_44 have equal integrals over the interval formula_45, using integration by substitution. But on the other hand, since formula_46, the sum of the two integrals is
Outer Circle railway line Outer Circle railway line The Outer Circle was built in 1891 as a steam-era suburban railway line, in Melbourne, Australia. It covered much of the modern City of Boroondara, including the suburbs of (from north to south) Kew East, Camberwell, Burwood, Ashburton, and Malvern East. At its longest stage, it ran from Fairfield station, on what is today the Hurstbridge line, to Oakleigh station, on the current Pakenham and Cranbourne lines. The Outer Circle railway was first advocated by a group known as the Upper Yarra Railway League in 1867 who suggested that the Gippsland Railway could be brought into