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
Straight-line grammar definition of the more general formalism of straight-line context-free tree grammars can be found in Lohrey et al. An SLG in Chomsky normal form is equivalent to a straight-line program. Straight-line grammar A straight-line grammar (sometimes abbreviated as SLG) is a formal grammar that generates exactly one string. Consequently, it does not branch (every non-terminal has only one associated production rule) nor loop (if non-terminal "A" appears in a derivation of "B", then "B" does not appear in a derivation of "A"). Straight-line grammars are widely used in the development of algorithms that execute directly on compressed structures (without prior