Area of a circle trigonometry, uses the coarea formula. Define a function formula_10 by formula_11. Note ρ is a Lipschitz function whose gradient is a unit vector formula_12 (almost everywhere). Let "D" be the disc formula_13 in formula_14. We will show that formula_15, where formula_16 is the two-dimensional Lebesgue measure in formula_14. We shall assume that the one-dimensional Hausdorff measure of the circle formula_18 is formula_19, the circumference of the circle of radius "r". (This can be taken as the definition of circumference.) Then, by the coarea formula, Similar to the onion proof outlined above, we could exploit calculus in a different way in
A Huge Ever Growing Pulsating Brain That Rules from the Centre of the Ultraworld A Huge Ever Growing Pulsating Brain That Rules from the Centre of the Ultraworld "A Huge Ever Growing Pulsating Brain That Rules from the Centre of the Ultraworld" is the debut single by the ambient house group The Orb. It was originally released in October 1989 and made the UK Singles Chart in 1990, peaking at #78. The 'Peel Session' version was also voted into #10 place in John Peel's 1990 Festive Fifty. In April 1991, it was released on the debut album "The Orb's Adventures Beyond the Ultraworld". The title is taken from a sound effects track from "Blake's