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### MAT1112_ch11_v2007

Course: MAT 2070, Spring 2011
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Word Count: 872

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11 Applications CHAPITRE de lintgrale multiple. e Ce chapitre sera tr`s bref. Il existe un grand nombre dapplications de lintgrale multiple. Il sut de e e penser aux notions desprance et de variance en probabilits ou encore des quations intgrales. Beaucoup e e e e de ces applications seront discutes dans dautres cours. Ici nous nnumrerons que quelques-unes, surtout e e e relies ` la physique. Plusieurs quantits...

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11 Applications CHAPITRE de lintgrale multiple. e Ce chapitre sera tr`s bref. Il existe un grand nombre dapplications de lintgrale multiple. Il sut de e e penser aux notions desprance et de variance en probabilits ou encore des quations intgrales. Beaucoup e e e e de ces applications seront discutes dans dautres cours. Ici nous nnumrerons que quelques-unes, surtout e e e relies ` la physique. Plusieurs quantits physiques peuvent tre exprimes comme des intgrales multiples. ea e e e e De tels expressions sont fondes sur la dnition de lintgrale comme la limite dune somme. e e e Si une quantit de mati`re est contenue dans une rgion R de R3 et (x, y, z ) est la densit par unit e e e e e e e de volume au point (x, y, z ), alors le centre de masse (x, y, z ) de celle-ci est dni au moyen dintgrales. En a physique, pour un syst`me de n particules, alors la composante x de ce centre par rapport ` laxe des x est e dnie par lquation e e m1 x1 + m2 x2 + . . . + mn xn m1 + m2 + . . . + mn dans laquelle mi est la masse et xi est la coordonne par rapport ` laxe des x de la position de la i-i`me e a e e c e e e u particule. On dnit y et z de la mme faon. Si la quantit de mati`re est distribue continment dans e la rgion R et (x, y, z ) est la densit au point (x, y, z ), alors nous sommes amens ` dnir son centre de e e eae masse (x, y, z ) par x= x (x, y, z ) dx dy dz , (x, y, z ) dx dy dz R R y (x, y, z ) dx dy dz (x, y, z ) dx dy dz R R y= et z = z (x, y, z ) dxdydz . (x, y, z ) dx dy dz R R Il est aussi possible dans la situation prcdente de dcrire le moment dinertie par rapport ` un axe. ee e a Nous nous limiterons ` dcrire ce moment par rapport ` laxe des z . En physique, le moment dinertie dun ae a syst`me de n particules par rapport ` un axe de rotation est dni par lquation e a e e 2 2 2 m1 r1 + m2 r2 + . . . + mn rn dans laquelle mi est la masse et ri est la distance laxe ` donn de la i-i`me particule. Si la quantit de a e e e mati`re est distribue continment dans la rgion R et (x, y, z ) est la densit au point (x, y, z ), alors le e e u e e moment dinertie I par rapport ` laxe des z comme axe de rotation sera a (x2 + y 2 ) (x, y, z ) dx dy dz. I= R Ce moment dinertie permet de dcrire lnergie cintique dun corps rigide qui tourne autour dun axe avec e e e une vitesse angulaire comme I 2 /2. Il y a une version 2-dimensionnelle du moment dinertie. Si R est une rgion de R2 et (x, y ) est la densit e e par unit daire au point (x, y ). Alors son moment dinertie par rapport ` laxe des x est R y 2 (x, y ) dx dy . e a Exercice 11.1: Soient la rgion R ` lintrieur du ttra`dre dans R3 dont les sommets sont (0, 0, 0), (1, 0, 0), (0, 2, 0) et e a e ee (0, 0, 1) et la fonction de densit (x, y, z ) = x + y + z . Dterminer le centre de masse de cette rgion. e e e 97 Exercice 11.2: Dterminer le moment dinertie par rapport ` laxe des z de la rgion R constitue des points de R3 tels que e a e e z 0, (x2 + y 2 + z 2 ) 1 et 3z 2 (x2 + y 2 ). Exercice 11.3: a) Dterminer le moment dinertie par rapport ` laxe des z de la rgion R constitue des points de R3 tels e a e e que 0 z H , (x2 + y 2 ) R2 . b) Dterminer le moment dinertie par rapport ` laxe des z de la rgion R constitue des points de R3 tels e a e e que H/2 y H/2, (x2 + z 2 ) R2 . Exercice 11.4: Soient a et b, deux nombres rels tels que 0 < a < b. Considrons le tore T engendr par la rotation du e e e cercle dquation (y b)2 + z 2 = a2 autour de laxe des z . Calculer la masse T (x, y, z ) dx dy dz de ce tore e en sachant que la densit est constante sur les cercles contenus dans des plans parall`les au plan des x, y et e e dont les centres sont situs sur laxe des z et quen un point A, elle est proportionnelle ` la distance de A au e a centre du mridien qui porte A. e 98
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CHAPITRE 12Intgrales impropres, fonctions gamma et bta et transforme de Laplace.eeeDans ce chapitre, nous revenons aux intgrales simples, mais cette fois soit lintervalle dintgration, soiteela fonction ` intgrer, soit les deux ne sont pas borns. T
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Kettering - MECH - 115
00 introductionmy expectationsyour expectationsgradingon-line quizzeswhat I expect from you: active learningbefore classread chapter before we beginwarm-up quiz (on-line) on reading*during classthink/pair/share quizzes*be prepared with question
Kettering - MECH - 115
01 Motiondiagramsuse movie to studymotionfixed time betweenframesdon't pan the camera!usually 1/30th of a secstrobe pictures arefasterfaster objectshave greater positionchange between framesWhich car is going faster, A or B? Assume there are
Kettering - MECH - 115
02 Velocity &amp;accelerationrv(average) velocity: tvector displacement rtime interval tdraw the velocity vector vvthe same size as the displacement vectordon't forget the difference in meaning, unitsvelocity:how fast it's goingAND in what dir
Kettering - MECH - 115
03 Motion in1 dimensioncoordinate systemoriginpositive direction: conventionx: right y: upposition-time graphsposition on vertical axis(dependent variable)even if position ishorizontaltime on horizontal axis(independent variable)can pick off
Kettering - MECH - 115
04 position, velocity and accelerationif v is constant s=v s tdisplacementchange in positionarea under v(t)workbook:2.2b page 2-3displacement when v isn't constanttfs f = si v s t dttiintegral = areaunder curvebetween curve andaxisintegrat
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08 Identifying forcesA force is. a push or a pullI push/pull on your handdo I exert a force? do you?does it matter if our hands move?I push/pull on a doorknobdo I exert a force? does the door?does it matter if the door moves?does a spring exert a
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