In modern math, this formula can easily be computed using calculus – it is, up to scaling, the integral Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying Cavalieri's principle – specifically, comparing the cone to a ( vertically scaled ) right square pyramid, which forms one third of a cube.

Without too much math we can write probability of choice statements based on attractiveness, and these take a form similar to some varieties of disaggregate demand models.

( Without local convexity this is false, as the spaces for < math > 0 < p < 1 </ math > have no nontrivial open convex subsets.

Without the presence of the king, the old institution of the Hoftag, the assembly of the realm's leading men, deteriorated.

Without his testimony, legal sources said the chances of convicting the three men were reduced.

Without a regular army and police force ( which in England was not established until 1829 ), it had been the duty of certain men to keep watch and ward at night and to confront and capture suspicious persons.

Without the benefits of modern training, men of the era were, in contrast, expected to run themselves to complete exhaustion during competitions.

Waggonner wrote: " Without men of courage and willingness to take up arms to defend America overseas, future generations of Americans would not be able to enjoy the liberties we cherish.

Without their skirmishers in front of them, the French column blundered into the 945 men of the 50th Regiment.

Without Archduke John's men, the Austrians could muster only 113, 500 infantry, 14, 600 cavalry and 414 guns for the second day of battle.

" There is no law nor lawyer's feesAll men are free from fury, For ev ' ry one do's what he please, Without a judge or jury: The summer-time is warm they say, The winter's ne ' er the colder, They have no landlords ' rent to payEach man is a free-holder.

Without adequate reconnaissance or planning, Won Gyun decided to attack with the entire naval force of Joseon at his disposal ; a fleet consisting of 150 warships operated by 30, 000 men that had been carefully assembled and trained by Admiral Yi.

Without their 2, 600 men, Washington realized he did not have the forces to attack Princeton and New Brunswick.

Without exchanging words, the two hike up to an amphitheatre-like hollow in which Rip discovers the source of previously-heard thunderous noises: there is a group of other ornately-dressed, silent, bearded men who are playing nine-pins.

Without any knowledge of the whereabouts of Duerst and Streiff, the former residents of Glarus left Baltimore searching for the two men who arrived before them and the land chosen for their new home.

" Without resolution, Thoreau used " his retreat to the woods as a way of framing a reflection on both what ails men and women in their contemporary condition and what might provide relief.

Without pack animals, Burke, Wills and King were unable to carry enough water to leave Cooper Creek and cross the Strzelecki Desert to Mount Hopeless, and so the three men were unable to leave the creek.

Without resources and men, Artigas withdrew to Paraguay in September 1820.

Adam's Curse: A Future Without Men ( also known as Adam's Curse: A Story of Sex, Genetics, and the Extinction of Men ) is a 2003 book by Oxford University human genetics professor Bryan Sykes expounding his hypothesis that with the declining sperm count in men and the continual atrophy of the Y chromosome, within 5, 000 generations ( approximately 125, 000 years ) the male of the human species will become extinct.

Without this political connection with Pompey, men dedicated to the Republic like Cato the Younger started a political campaign against Caesar, arousing suspicion and accusing him of wanting to overthrow the Republic and become King of Rome.

Without a complete documentation of the founders of the fraternity, the national organization adopted both of these men as the " founders " given their lifelong commitment to the fraternity and their service as visionaries for the development of the fraternity's ritual and national expansion.

Without informing Howe, St. Laurent contacted Drew, and the two men agreed that the minister's powers would expire in 1959 unless sooner renewed.

Without consulting the Connecticut colonists, Massachusetts, in August, sent a punitive expedition of ninety men under John Endicott to Block Island with instructions to kill every Niantic warrior and capture the woman and children, who would be valuable as slaves.

Without these the weapons could be highly dangerous: Hakluyt's " Voyages " records the death of one of the men on Cavendish's circumnavigation in the 1580s due to an accidental discharge during a hurried re-embarkation on the coast of Ecuador.

Without much conversation following the unusual incident which just took place, the men went to sleep.

Arya Stark and her friends encounter a group of men known as the Brotherhood Without Banners, led by Lord Beric Dondarrion and the red priest Thoros of Myr.

" Without the snag ," a state investigator said, " those two men wouldn't be with us today.

Without really changing the general subject, I take this opportunity to confess that I am troubled by doubts, not only about pacifism, but also when asked to join in the protest against a law that most of those who consider themselves humane and liberal seem to regard as obviously barbarous ; ;

Without that knowledge, without that capacity to think, you can easily become a victim of people who seek to take advantage of you ".

Guided by Arkyn, Corum, Rhalina and companion Jhary-a-Conel cross the planes and encounter the King Without A Country, the last of his people who in turn is seeking the City in the Pyramid.

Without any effort, Munch had a steady stream of female models, some of whom he may have had sexual relations with, and who were the subjects of numerous nude paintings.

The Kingpin discovered his plan and threatened to kill the hero, but he was rescued by the Man Without Fear, Daredevil, who then recruited him as a part of his team.

" Without Chains " is about Murat Kurnaz, a Turkish citizen who was born and raised in Germany, held at Guantanamo Bay detainment camp for four years.

Other artists who have claimed Roxy Music as an influence include Steve Jones and Paul Cook of The Sex Pistols, Siouxsie and the Banshees, The Cars, Grace Jones, Kate Bush, Adam Ant, The Human League, Japan, Duran Duran, Simple Minds, ABC, Spandau Ballet, The Fixx, Depeche Mode, Men Without Hats, Nile Rodgers, Annie Lennox, Morrissey, Jarvis Cocker and Neil Hannon.

Without hesitation, Truman picked Marshall, adding " I don't think in this age in which I have lived, that there has been a man who has been a greater administrator ; a man with a knowledge of military affairs equal to General Marshall.

" Without looking to see who is speaking, Butterfly corrects him, " Madam Pinkerton, please.

Without mentioning names, he made pointed allusions to tyrants who usurped the freedom of the people, and he excoriated their allies, the rich and powerful who neglected and exploited the poor.

He did so based upon the < span lang =" fr "> Bourbon </ span > response to the " Year Without a Summer " in 1816, during which < span lang =" fr "> Louis XVIII </ span > relaxed tariffs during a series of famines, caused a downturn in prices, and incurred the ire of wealthy landowners, who were the traditional source of < span lang =" fr "> Bourbon </ span > legitimacy.

* Without a declaration of hostilities, Greek statesman, Aratus of Sicyon, who has gradually built up the Achaean League into a major power in Greece, makes a surprise attack on Corinth and forces the withdrawal of the Macedonian occupation troops.

In The Musical Quarterly, Martial Teneo and Theodore Baker wrote, " Without the example set by Hervé, Offenbach might perhaps never have become the musician who penned Orphée aux Enfers, La belle Hélène, and so many other triumphant works.

Without any official licensing practices, there were no ethical or professional limits on who could practice NLP.

On one level Act Without Words I “ seems a behaviourist experiment within a classical myth ”, that of Tantalus, who stood in a pool of water which receded every time he bent to drink it, and stood under a fruit tree which raised its branches every time he reached for food.

Without a DNA sample, it is not possible to reconstruct the complete genetic makeup ( genome ) of any individual who died very long ago.

Hood ( Confederate ); Ulysses S. Grant ( Union ) was named to Fort Jones, but was Absent Without Leave for whatever his tenure would have been ; George Crook ( Union ), who would arguable become one of the greatest leaders for the Grand Army of the Republic less than a decade later ; and George Pickett ( Confederate ).

The tag line of the commercials was " Without sports, who would cheer for the Nimrods?

Valley is the hometown of Gail Rock, who used it as the basis for the small town Clear Rock in her Addie Mills books, of which " The House Without a Christmas Tree " is best known.

Without a consistent, strong international focus on the realising the rights of farmers who conserve and sustainably use PGRFA to save, use, exchange and sell seeds saved on-farm, genetic variety of crops and related agricultural biodiversity will suffer.

Guest, who in 1948 published the book " Gardening Without Digging ".

In some of his other roles during the 1960s and 1970s, Bogarde played opposite renowned stars, yet several of the films were of uneven quality, down to demands or limitations set by the studio or their scripts: The Angel Wore Red ( 1960 ), playing an unfrocked priest who falls in love with cabaret entertainer Ava Gardner during the Spanish Civil War ; Song Without End ( 1960 ), as Hungarian composer and virtuoso pianist Franz Liszt, a flawed film made under the initial direction of Charles Vidor ( who died during shooting ), and completed by Bogarde's friend George Cukor, the actor's only disappointing foray into Hollywood ; the campy The Singer Not the Song ( 1961 ), as a Mexican bandit co-starring John Mills as a priest ; H. M. S.