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
What to the Slave Is the Fourth of July? What to the Slave Is the Fourth of July? "What to the Slave Is the Fourth of July?" is the title now given to a speech by Frederick Douglass delivered on July 5, 1852, in Corinthian Hall, Rochester, New York, addressing the Rochester Ladies' Anti-Slavery Society. The speech is perhaps the most widely known of all of Frederick Douglass' writings save his autobiographies. Many copies of one section of it, beginning in para. 32, have been circulated online. Due to this and the variant titles given to it in various places, and the fact that it is called a July