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
Fife Circle Line line from Inverkeithing that loops back round to Kirkcaldy by an inland route via Cowdenbeath through the old Fife coalfield. Narrowly speaking, just this line could be called the Fife Circle. The current service is actually a combination of two previously separate local routes - Edinburgh to Kirkcaldy & Edinburgh to & . During the 1970s & 80s British Rail only ran a regular daytime service on the Dunfermline line as far as Cowdenbeath; & Cardenden were only served during the weekday business peaks (as can be seen from Table 242 of the UK All Line timetable of that era),