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
Circle Line Sightseeing Cruises location. In 1962 it bought the famous and venerable Hudson River Day Line, keeping it as a separate brand until 1985. In 1981, the Circle Line split into two companies—Circle Line Sightseeing Cruises and Circle Line Downtown. In 1988 the 42nd Street company bought World Yacht's operating upscale dining cruises from Chelsea Piers - currently World Yacht Dining Cruises. In 1998 the 42nd Street company also launched The Beast, a speedboat ride which takes tourists around the Statue of Liberty and goes 45 mph. In 2007, the United States National Park Service said it was going to terminate Circle Line