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# Integral du over under betting

Cross out that row, that column, 1 times 1 minus 0. So it's going to be plus k. So if we want the magnitude of this, this whole thing right over here is just going to be the square root-- I'm just taking the magnitude of this part right over here, the actual cross product.

It's going to be negative 1 squared, which is just 1, plus negative 2v squared, which is 4v squared, plus 1 squared, which is just 1. So this whole thing is going to evaluate to-- we have 2 plus 2v squared du dv. Or actually, let me-- I almost made a mistake. That would have been a disaster. And if we want, maybe it'll help us a little bit if we factor out a 2 right over here. This is the same thing as 2 times 1 plus 2v squared du dv. If you factor out the 2, you get this is equal to the square root of 2 times the square root of 1 plus 2v squared du dv.

And I now think we are ready to evaluate the surface integral. So let's do it. All right. So let me just write this thing down here. So I'm going to write everything that has to do with v, I'm going to write in purple. So I'm just going to write the ds part right over here. It is the square root of 2 times the square root of 1 plus 2v squared. And then we're going to have du, which I'll write in green, du, and then dv. So this is just the ds part. This is just the ds. Now, we have the y right over here.

All right, so I'll write that in purple. So y is just equal to v. That is y-- let me make it very clear-- and all of this is ds. And now I can write the bounds in terms of u and v. And so the u part, u is the same thing as x. It goes between 0 and 1. And then v, v is the same thing as y, and y goes between, or v goes between, 0 and 2. And I now think we're ready to evaluate.

And the u and v variables are not so mixed up, so we can actually separate out these two integrals, make this a product of two single integrals. The first thing, if we look at it with respect to du, all this stuff in purple is just a constant with respect to u, so we can take it out of the du integral.

We can take all this purple stuff out of this du integral. And so this double integral simplifies to the integral from 0 to 2 of-- I'll write it as the square root of 2v times the square root of 1 plus 2v squared. So I factored out all of this stuff. And then you have times the integral from 0 to 1 du, and then times d, and then you have dv.

And now, if this was really complicated, I could say, OK, this is just going to be a function of u. It's a constant with respect to v. And you could factor this whole thing out and separate the integrals. But this is even easier. This integral is just going to evaluate to 1. So this whole thing just evaluates to 1.

And so we've simplified this into one single integral. So this simplifies to-- and I could even take the square root of 2 out front-- the square root of 2 times the integral from v is equal to 0 to v is equal to 2 of v times the square root of 1 plus 2v squared dv. And so we really are in the home stretch of evaluating this surface integral. And here, this is basic. This is actually a little bit of u substitution that we can do in our head.

If you have a function, or kind of this embedded function right over here, 1 plus 2v squared, what's the derivative of 1 plus 2v squared? Well, it would be 4v. And we have something almost 4v here. We can make this 4v by multiplying it by 4 here, and then dividing it by 4 out here. This doesn't change the value of the integral.

And so now this part right over here, it's pretty straightforward to take the antiderivative. The antiderivative of this is going to be-- we have this embedded function's derivative right over here, so we can kind of just treat it like an x or a v. So this is the antiderivative of that. And then, of course, you still have all of this stuff out front-- square root of 2 over 4 And we are going to evaluate this from 0 to 2. We don't have to worry about that at 2 and 0.

And actually, this will cancel out. That becomes a 1. That becomes a 2. And now if you evaluate this at 2, you have 2v squared. So it's going to be equal to And then minus this thing evaluated at 0. Well, this evaluated at 0 is just going to be 1. Bronshtein K. Primitive function. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, log in to check access. In certain cases the domain of definition of a primitive function F x is wider than that of the given function f x. If the function f x is defined in a connected domain except for isolated points of discontinuity x 1 , x 2 ,…, x n , then the domain of definition of its primitive function may contain these points see p. Google Scholar. See footnote on p. For tables of elliptic integrals see p.

For the graphical integration construction of the graph of a primitive function from that of the given function see pp. The numbers A , B ,…, L can also be obtained by the method of undetermined coefficients see p. The polynomials P 1 x and P 2 x can be found easily, if the factors of P x are known, i.

In cases, when the integrals A are expressible in terms of elementary functions, they are called pseudo elliptic integrals. The symbol R denotes a rational function of the expressions to which it refers. Integrals of functions involving sin x and cos x together with the hyperbolic functions and the function e ax are given on pp.

In this case, Li x should be understood to be the principal value of the improper integral see p. The integral logarithm is related to the integral exponential function see p. The concept of the definite integral can also be generalized for functions defined in an arbitrary connected interval an open interval, an interval open on one side, a half-axis or the whole number axis or in a connected interval except a finite number of separate points.

The integrals considered in this more general sense belong to the improper integrals see p. The definite integral exists also for any bounded function with a finite number of points of discontinuity in the interval [ a , b ]. A function for which the definite integral exists in the given interval is called integrable in this interval. In order to avoid misunderstanding, the variable of integration has been denoted by t see p.

In a general case, when the direction of the force does not coincide with the direction of the motion, the work can be calculated as a line integral see p. For the Gamma function see p.

### FOOTBALL BETTING 90 MINUTE RULE

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Then she had a little weakness for exaggeration, for unconscious egotism in conversation, and I itched to tell her so. At the last moment the camera lost him again and pulled wide just in time to reveal the base of the mountain and the crowds of people on the embankment recoiling in shock from where the body had hit the ground. Had you a long acquaintance with her.

That sort always run into trouble sooner or later. The most I would say is that she met somebody the rest of us did not know, went swimming with him and that he dared her or enticed her into swimming when the tide was going out, a thing which, left to herself, she never would have done. What I find difficult to understand is why either of them bathed on an outgoing tide at all. I distinctly heard you tell Inspector Durrand, and prove it by the mask, that it belonged to another woman.

Liv swept the baseball cap from her head and stared back, doing her best to maintain a neutral expression. She could feel his scrutiny crawling over her face, like the feelers of some giant insect. The blood sang in her ears and she was sweating from a combination of stress and poor air-conditioning.

His eyes continued to slide over her face, then dropped down to roam over her body. Ordinarily, Liv would have been outraged by this, but now she felt relief. Nor had Jofre ever heard of anyone possessing an artifact such as he had found. And when he thought of that he knew a trickle of cold within. He lowered the bottle and blew out a gusty breath of appreciation. This hotel is proud of its cooking.

Want to sample the superb treat. Waking from the dream, she reached out for Gabriel, as she had in the cave, expecting her hand to find his solid form, but he was not there. It was occupied by John Mann, resting at last. She could tell that he was dead, but she could also feel his peace. Slipping from her bed, she moved across to stand beside him, laying her hand on his. She felt anxious for Gabriel, wondering where he was and why he was not sitting vigil beside his father, whom he had found and lost again in the space of a few hours.

The building seemed deserted, all sounds coming from the compound beyond. He did not want to die, but he could accept it if he had to. Perhaps that was why it took him so long to realize the true situation. There was a clip lock on it: it was easy to open. But it were them fillum folks ordered it. Wanted to make a picture of the North Pole explorers, or summat of that, so they tell me.

The ice all throwed out in a pond they dug in the garden. It were over there where they planted them bits of pine trees. You just need them to switch their mobile on. She watched the water turn from grey to clear as it sluiced off her body and spiralled down the drain, carrying away the grime of the night. Her hand continued up her side and down her arm to where a cross-hatch of smaller scars corrugated her skin, scores of thin lines scratched during a childhood troubled by the lack of a mother and a sense she was a stranger in her own family.

If only her father had told her back then what she had discovered for herself on that shady porch in Paradise, West Virginia. Convert from to. Then , so. Rewrite using and. Tap for more steps Let. Tap for more steps Rewrite. Divide by. Rewrite the problem using and. Combine and. Since is constant with respect to , move out of the integral. Pretend to be asleep except not in the library again Canned response: "As with any function, the integral of sine is the area under its curve. She was thirteen, and grew more beautiful each year.

She had the same elfin beauty as her mother, in contrast to her much taller older sister, who smiled proudly at her through the grille, and brushed her cheek with a kiss. Her mother had told her as much in a letter. There was a boy she had a crush on at school, and who liked her. I have to assure them that we are not facing some kind of moral crusade.

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It was when I was about to leave the scene of these small disappointments that my one really important feat was performed. As I was passing out of the gate into the light of freedom I suddenly turned and looking the warden gravely in the eye, soon had him in control. Gilstrap and the diversion of nearly all the travel to another road occurred so nearly at the same time that no one has ever been able to say which was cause and which effect. Beyond the park, however, the marshes were as Nature had intended them to be.

Palgrave walked a couple of miles back along the road towards Saltacres and then took a path which led seawards, but at the end of a mile or so the path petered out and he found his way blocked by a creek which meandered through the marshes and was too wide to jump across. He spent the day quietly, thinking about his book, and on the following morning he was in the bar finishing his first pint when Adrian walked in.

Palgrave had told him where he would be staying and had been expecting that he and Miranda would at least telephone him before they returned to London, but he had not thought to see either of them because they had no transport. Adrian, however, had not come merely to see him, but to make a pertinent enquiry.

I should soon give her the bird if she showed any signs of trying to team up. They were more ghastly, these shadowy people, than though they had stepped forth in their coloured coats to speak. Something was falling, drop by drop, slowly. I wanted light, and the knowledge of human presence in this choking stuffiness of wax and wigs. Bencolin and I would have a good laugh at it, over brandy and cigarettes, when we had left this evil place.

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What was the use of talking about a thing like this with his only real solid friend: Sammy the Black. Her right hand toyed with the curly hairs on his chest. But their plans to tour a museum or two and eat lunch in the city had fallen prey to deeper, more passionate needs. And every hour she spent in his company helped her push away the dark memories of the carnage at Temple Emet.

Reacting to the sudden tension in her voice, he was already up and getting dressed. He saw her urgent gesture and switched the set on. A fortnight is the average in the summer, a week in Spring or at this time of year. I left the office at six last evening when everything was still in order and came in at nine this morning as usual.

I did not take my records out of the filing-cabinet immediately, as I had some odds and ends of correspondence to clear up and that involved nothing but opening my desk. Detective-Constable March will check it for fingerprints. Much obliged for your help in that little matter, sir. It admitted that the crashed planes had carried nuclear armaments but said nothing about radioactive contamination or missing bombs.

The ground was covered with sawdust, and it was in the air with the smell of wood and sap, and there was the constant, irregular sound of friction, banging, and someone whistling half under his breath. The tide slurped on the stone of the slipway. One hand rested idly between the lapels of his coat, as if he were adjusting his tie. Butlers are traditionally large and overwhelming, but they are apt to run large in the wrong places. This butler was large in the right places.

At four-day-week norm, allowing for projected overtime as against health losses, three million eight hundred thousand dollars per year. Colonel Eichmann threw every possible resource into finishing his mission of genocide.

Each day in new uprisings the Sonderkommandos snatched SS guards and their dogs and threw them into the crematoriums. At last every Sonderkommando was executed and a call went out for a new group from Auschwitz. Twenty thousand Jews, the cream of Jewry, who had been under guaranteed protection at the Czech camp of Theresienstadt, were ordered transferred to Birkenau for extermination.

He and other Sonderkommandos stood by until the shrieks of agony and the frantic pounding on the iron doors stopped. They waited another fifteen minutes for the gas to clear. And there had been failures, some of them quite harsh. In his youth he had been able to stand in the Cordeliers with his eyes closed and imagine that if he opened them he would see the ghosts of Robespierre, the giant Danton, and the raving Marat, hear the rattle of tumbrels over the stones, and smell the fear.

That had been a mistake that had nearly cost him his life. He had felt a starry-eyed pity for her that had bordered on love. He would not get a second chance at this if something went wrong. It was a smaller camera and not as sophisticated as those of the other networks, but ECNS was supposed to be a new service one based in Eastern Europe. Nobody asked for an explanation, but Halovic wanted his cover story ready if anyone did.

The only thing that mattered was helping their friend. The old rancher would choose to stand and fight. Dark cattle pockmarked the green as far as he could see. He wanted to return with his own boat: a sleek thirty-footer and he would charter her for fishing: going along as Captain with one crew-someone like Sammy: even Sammy.

Like an aching tooth, the dream of owning his own boat, feeling the surge of the sea nagged him for as long as he could remember and was nagging him now as he sat at the window. There was no harm in looking at it, was there. Apply the sine double-angle identity. Multiply by. Multiply the argument by. Combine fractions. Tap for more steps Combine. Simplify denominator. Tap for more steps Apply the … borderlands 3 the slaughter shaft For some reason, I felt the urge to speak quietly.

In the distance, I thought I saw a clearing with a few clumps of mud brick sticking above the grass like rotten teeth. He crouched in the grass, trying to lower his profile. Then he glanced back nervously at Philip of Macedonia. Philip shrank back to a small wax statuette. Walt pocketed our croc, and we began sneaking toward the ruins. Jerry fought to control his irritation. I wanted to do it while they were still fresh in my mind.

His anger flashed into full bloom. When an angel wakes one of us dies. The second is justification of herself by accusation of you. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral. I was about to sign the deal with the theatre.

She asked me to sign half of my holdings in Three Hundred International over to her. If I did not agree to do so, she said she would go to the press, tell everyone that she was still legally my wife, that I had conspired to have her killed and that she had survived the attempt. Me, who had only ever loved her. I could not allow an ugly court case, just when I was fighting to make a name for myself in London.

And he had pinned so many hopes and plans on her special abilities. By remaining here close to Thallo, at least she had a chance of averting the disaster. This, she knew, would be more difficult than all of her previous vigorous exercises. His chair was studded with stray hunter-seekers, and each projectile he grabbed was armed with a different poison. His hands, arms, chest, and back had been stung repeatedly.

He could barely concentrate now, forced to devote most of his effort to counteracting the dangerous chemicals building up in his bloodstream. Bodies lay in heaps on the dais and out in the audience. The crowd was a cacophony of screams. This question is a good candidate for the integration by parts method, as it is the product of two different parts.

Petrov and his men are on borrowed time. Whenever possible, they took the girls with them. While Horst and Ulm were there, they spent as much time as they could enjoying their company. Ulm was leaving for the front when he got back, and Horst was stationed at divisional headquarters in Munich, which he seemed to find vastly amusing.

He and Brigitte had much more in common with each other than either of them did with Beata. Whatever it was that had taken hold of her had pulled her feet between the staves of the trellis, so that her ankles were resting on the crossbars. She was alone now, left in this graceless position, trying to imagine a way out of an absurd predicament.

This is my worst fear made real. My mother alone in her kitchen, trying to call for the maid, attempting to reach the telephone in the hall. No one in London should ever be surprised by the weather, but this year we can expect winter to arrive with a vengeance. Besides, he had something else on his mind. He was still annoyed about his hotel room.

But there was no way he could pursue the matter further. Besides, as long as energy could be kept expensive to the consumer while industry got it cheaply by tax relief, one could make a profit selling an endless array of windmills, solar panels, storage batteries, and alcohol stills to two hundred and fifty million Americans. This view accorded well with groups that - with some justification - yearned for an earlier time.

The tenets of Mormonism, outwardly simple in its absolutes and its demands that each Mormon household be as independent as possible, became more attractive. They comprised a heaven-sent bloc of patriots, and a hell of a bloc of votes.

I tested him, since the draug carry diseases at times, but it appears he is clear of that, which is a lucky thing. All he needs is blood for now, and rest. He should be better very soon, I promise you. Today that served him very well, but you must understand that he will require more than physical strength. I do not know which Shane experienced. Eve would have been his listener, but there was a gap between them now that they had to shout across. Dec 04, curl: 52 empty reply from server It would be slow, but once it got started, if the news were important enough, the people would hear it.

Then, after he leaves, I want to see my wife and the children. After that, the chief of the Secret Service detail here. The ground had a gray cast to it. It could mean only one thing, but he refused to let his mind dwell on it. In the eyes of those I return to I may be considered one who has failed. And the Holder will soon be, judging by the attitude of these Tssekians we have just seen.

Also-were you oathed directly to him or to his subordinate. That, too, was part of issha training, but there was no reason for her to use her knife on herself in this situation-the oath was not a full service one he was sure. For Jofre it was a far more comfortable one, even though their quarters were crowded.

He had hoped to establish some closer communication with the Jewelbright. Integrals of Trigonometric Functions: When it comes to finding the integrals of trigonometric functions, we often need to use a number of different trigonometric identities and girl farts in mouth I stopped, trembling, and stared at him for a few long, horrible seconds. He just opened his arms, and I fell into them, sobbing my heart out as he stroked my hair.

It was healing, but the red line down the middle told him that doctors had removed the stitch too soon. The other occasionally pulled the other string, controlling the drinking hand of the dummy. The motorboat could no longer be heard.

They made another fifty yards, and Gigi could see the pontoon and the other boats in the dim light. Larry reached over and switched off the ignition. As he spoke, they both saw the beam of a flashlight on the trees above the pontoon, then it went off. They were now drifting backward with the current, and the boat began to turn sideways.

That struck a sudden spark in his brain. He had to collect a few of his friends. If they moved fast before the phones came back on, they could really score. Reaching under the seat, he pulled out a 9mm automatic. There were about seven or eight she found likely, and another half dozen possible, though she believed they lacked the courage, or the indifference to values of modesty or loyalty, to have taken such a step. And all the ordinary moments in between.

But I missed you so much, every single day. In the end, it was like a death. Papa must have returned them without showing them to me. Share His face was grim, eyes hard and steady, but there was an emotion in them that Rathbone found chilling. Perhaps he was trying to get Parfitt to stop blackmailing someone.

Camilla lay down beside him, propped her chin on her fists and gazed out towards the pebble-ridge and the sea. Palgrave frowned at the blank page of his notebook. Valentine took it as a yes, and entered the study. He saw Mink put his hands onto the desk, and ball them into fists. My question is, how does one go about evaluating this, since its existence seems fairly intuitive, while its solution, at least to me, does not seem particularly obvious. There, as the lowest-ranking officer, Angie read incoming cables and updated battle maps with colored pushpins.

Sometimes he stood at the maps with a pointer as generals discussed battle plans. He remained in the Army for five years, retiring with the rank of major. In , he was conducting an auction at a golf tournament. He felt close to the truth and wanted to talk to Andreas Renalda, but nobody knew his whereabouts. The day was grey and dull, the skies louring with the threat of rain.

As he reached out again she kicked back with her spiked heel, connecting hard with his chest. The chill stone balustrade curved away on either side. She intended to follow the path around, knowing that she would be able to see if he changed direction. There was no time to put any distance between them.

Bryant was aware of other people every second of his life. What women thought of him mattered to a punitive degree. It was to do with being young, of course. My whole body was crawling with the horror of the situation.

I controlled myself, forced myself to lock the trunk, then the garage doors and then I walked quickly back to the bungalow. The strain was showing on her face. It seemed to me she was older, thinner and very tense. I took out my handkerchief and wiped my face. The breeze does rise and fall a little, but not enough to matter. He started to say something else, but then a familiar voice knocked the words out of his head. She was trotting up the slope from the village. He had expected to spend the rest of his life here, and had tried to ensure that it would be as pleasant as possible - though Arvagan had warned him that wizardry had no guarantees.

Hanner sank down onto it gratefully. He lay back and closed his eyes. If anything might cause Christina to lose her self-mastery, her judgment, it would be Charlotte flirting with the general-and even perhaps with Alan Ross. They were just late enough still to be civil and yet also to cause a pleasing stir of appreciation among the guests already thronging the hall.

She looked like the spirit of an elusive early English summer, when the blossom is still clean and the air dappled with cool and shifting light. I am compelled to pronounce it tobacco smoke. But now that I look again, you are not entirely wrong: there is a considerable admixture of jute, moss, and I think tallow.

For a few seconds Gail stood in the doorway of the familiar bedroom, watching as he snored softly. To her surprise the place was clean. His clothes were neatly folded over a chair, and what she could see of him was scrubbed and clean shaven. It was an irritant she had learned to live with, as she ignored his puttering around until she later joined him for breakfast.

Oct 13, Dec 20, laguna hills civic center No one but the original Navigator can find them, Sire. They were expecting only a brief passage. But he had seen enough excesses and would not add to them unnecessarily. I fear I have trampled the feelings of those people.

The Count patted Marie on her blonde head and handed the girl a dagger. The muscular, fully grown soldiers faced Marie, looking at the girl in uneasy confusion as the door sealed, leaving the observers outside. She will fight in complete darkness. You have already proved your flaws and your guilt.

Wensicia, you have made a bad choice of husbands. Therefore, I see little point in keeping him around. He dawdled down the street, wondering why Lance would be in a pub before lunch. No more than a minute had passed when he saw two people get out of a parked saloon car and head for the pub. They were Mediterranean in appearance, and the woman was quite beautiful. Stone tucked the newspaper under his arm and crossed the street. He saw the three of them seated at a corner table, and he moved around to the side of the building for a better view.

Then, confusing him, she was crying. A moment later she was suggesting that he come to her place and pat her shoulder - among other things. He still talks to me, God knows why. The horse jumped ahead, through the opening in the guardrail and into the water.

Sarah was washed in a wave of ice-cold spray that made her sjiiver. Sam had jumped for it, but she lost sight of him in the next instant. He struggled back to his feet, nursing a bleeding wrist. When I knock you down you will make one big crash. It will make louder and sweeter music than your Battleville band ever played. Flatcars loaded with cannon and cases of ammunition were interspersed with boxcars full of Union soldiers.

The hole was directly over his heart. His eyes, when the lids were withdrawn, had rolled up, showing only the whites. Even when they had finally gone to bed, she had cried in the dark. When he touched her, her face had been wet with tears. What Blantyre must be feeling was a devastation imaginable only to those who had experienced it. Stoker came in almost on his heels with the brandy, and Blantyre accepted it.

You might want to consider how the succession for the chairmanship will work. When they cleared the rooftop of the mansion Kirsha paused. The surrounding buildings turned until only the roofs were visible, and then shrank down to the size of floor tiles. Koo pastes a happy smile on his face. But the program itself had been such a catalogue of horrors that Koo-and probably everybody else watching it-had promptly forgotten that quick enigma.

An area code, and a phone number done as a name. Gabriel had offered to go, but with normality returning to the city, and too many people on the lookout for him, she had gone instead, thrilling at the opportunity to play a small part in something as momentous as this. The walls behind them were painted with an elaborate and vividly imagined mediaeval fresco depicting the day of reckoning. Dr Anata wondered if they would let her jump the queue if they knew she was here to try to avert the very thing they were all staring at.

It took a further twenty minutes before she made the short walk of shame and closed the curtain behind her. She perched on a wooden ledge, bringing her face level with the grille. As an academic she was wholly unused to situations that placed her at risk and her mind was now in overdrive, imagining all sorts of scenarios involving security guards and brutal interrogation. Kings fall, battles are lost or won.

Or maybe it was just all that white. Walking down the sterile halls he knew he would never get accustomed to the smell. Why, in the high-tech twenty-first century, did hospitals still have to smell like alcohol. Did they even use alcohol anymore. Maybe the odor was all in his head. Then why were you so quick to tell him to jump in the water.

Witness: Laurie said something to him about how the bathing suit was expensive and Jacob should take better care of his things. It was bad enough that for the first time in nearly ninety years someone had almost escaped from the Citadel. That he had perished in the process was the only bright spot on the horizon. There was also no getting away from the fact that it was ultimately his responsibility.

Traditionally the Citadel guards would have been skilled with crossbow, sword and dagger, but times had changed. Now a wrist holster containing a Beretta 92 double-action pistol with a full clip loaded with parabellums nestled within the loose sleeves of their russet red cassocks. The guard heaved open the door to let him pass. And yet the knots had been peculiar, identical to those on that terrible parcel in the churchyard and all the others.

If he sent her back to Tortoise Lane would she tell Mrs. Mapes what he had asked her, and where she had taken him. He did not hold much hope that she would hold out against Mrs. Nellie lived in a cocoon of hunger and fear. Tortoise Lane was her home-probably all she knew. Voskavich left then and Rozhdestvenskiy stood amid the bodies, by the sealed doors, studying the faces.

They had died for something important-what was it. Pulling it open toward him, he shone his light inside. He felt like Carter at the discovery of Tutankhamen. No golden idols were here, but file cabinets, unopened, unlike the ones in other parts of the complex. There was no pile of charred papers and microfilm rolls in the center of the floor. There was community responsibility for health, welfare, and education.

What of those who were disgruntled over their assignments. What of those who objected to the cooking or to living in such tight quarters. What of the clash of personalities. The captain did not turn around as the journalist entered. Her thin, bony face was expressionless. She looked around the deserted bridge. I may be uneasy about it, but I intend to have the job done right.

A Tudor named Henry the Tenth on the English throne, and a North American federation of Inca chiefdoms with not much technology beyond good roads and the telegraph. And then I remembered another thing, which caused me very heartily to curse myself. It came from the Marat tableau all the time.

Now all that money meant nothing. He heard shots behind him and felt a slug rip past his ear. It seemed to pull him along and he ran faster. Another bullet gouged splinters off a tree in front of him. Augusta had known her too long to be deceived by the arrogant stance, the squared shoulders under the thick satin. Good heavens-she was still so much of a child, as feckless as a summer day. She saw what she wanted and reached for it.

Where on earth had she come by such abandon. It was certainly not her father. But it was not his place to ask questions. Did that come from the shrinking of the planet-born who had never been in space, or was it a cautionary impulse triggered by something else. There were a number of attendants around but none of them showed the characteristic features of the Asborgan-born. These were mainly off-worlders and some were truly alien.

In this very mixed group he might not have attracted the general eye, for he was wearing the livery of a high lowland house and accompanying a young Highblood. Surface integral example part 2. Surface integral example part 3: The home stretch.

Surface integral ex2 part 1. Surface integral ex2 part 2. Surface integral ex3 part 1. Surface integral ex3 part 2. Surface integral ex3 part 3. Surface integral ex3 part 4. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript Now that we've set up the parametrization, let's try to evaluate the integral. And the next thing we'll do is essentially try to express ds right over here in terms of du and dv. And we've seen this before.

So first, let's take the cross product, and we'll do that with a 3 by 3 matrix. I'll do that right over here. We set up, and I'll just kind of fill in what r sub u and r sub v is in the actual determinant right over here. So first we have our components, i, j, and k.

And now first, let's think about what r sub u is, the partial of r with respect to u. Well, its i component is going to be 1. The partial of u with respect to u is just 1, so its i component is 1. Its j component is going to be 0, partial of v with respect to u is 0. This should be parentheses around here. Partial of this with respect to u is just going to be 1 again. The partial of v squared with respect to u is just 0. So this is just 1 again. And then r sub v, the partial of r with respect to v, the i component is going to be 0, j component is going to be 1, and the partial of u plus v squared with respect to v is going to be 2v.

So it's a pretty straightforward determinant, so let's try to evaluate it. So the i component, it's going to be i. So we cross out this column, this row. It's going to be 0 times 2v minus 1 times 1. So essentially, it's just going to be negative 1 times i. So we're going to have negative i, so this is equal to negative i. And then the j component-- and we're going to have to put a negative out front, because, remember, we do that checkerboard pattern.

So cross out that row, that column. Let me make sure I got that. So it's just going to be 2v. But since it was the j component, which is going to be negative, it's going to be negative 2vj. Let me make sure I did that right. That column, that row, 1 times 2v is 2v minus 0 is 2v. The checkerboard pattern, you'd have a negative j. So you have negative 2vj. And then we have find the k component. Cross out that row, that column, 1 times 1 minus 0.

So it's going to be plus k. So if we want the magnitude of this, this whole thing right over here is just going to be the square root-- I'm just taking the magnitude of this part right over here, the actual cross product. It's going to be negative 1 squared, which is just 1, plus negative 2v squared, which is 4v squared, plus 1 squared, which is just 1.

So this whole thing is going to evaluate to-- we have 2 plus 2v squared du dv. Or actually, let me-- I almost made a mistake. That would have been a disaster. And if we want, maybe it'll help us a little bit if we factor out a 2 right over here. This is the same thing as 2 times 1 plus 2v squared du dv. If you factor out the 2, you get this is equal to the square root of 2 times the square root of 1 plus 2v squared du dv. And I now think we are ready to evaluate the surface integral.

So let's do it. All right. So let me just write this thing down here. So I'm going to write everything that has to do with v, I'm going to write in purple. So I'm just going to write the ds part right over here. It is the square root of 2 times the square root of 1 plus 2v squared. And then we're going to have du, which I'll write in green, du, and then dv. So this is just the ds part. This is just the ds.

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