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
The Penguin Goes Straight jewels, but they are caught red-handed by the Penguin and his agents and charged with burglary... The Penguin Goes Straight The Penguin Goes Straight is the 21st episode of the "Batman" television series, first airing on ABC March 23, 1966. It guest starred Burgess Meredith as The Penguin. Attending a matinee performance at a Gotham City theatre, the Penguin foils a thief’s attempts to steal a ruby from the beautiful actress Sophia Starr. Then, at the Millionaire's Club, he stops two crooks from kidnapping millionaire Reggie Rich from a steam room. When Batman and Robin arrive, the Penguin announces his