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Prévia do material em texto

JSFH
 
 
 Teste 2009.2:
 
 
 
 Quest\303\243o 1:
 
 
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curvas 1 e 3 s\303\243o a mesma curva e as curvas 2 e 4 tamb\303\251m s\303\243o as mesmas curvas, mas eu s\303\263 mudei o intervalo, j\303\241 que a interse\303\247\303\243o de 3cos(t)=1+cos(t) e isso nos leva a cos (t) =1/2, ou seja, t = + ou - (Pi/3) + 2*K*Pi, sendo K um valor inteiro qualquer. O desenho acima corresponde a regi\303\243o R.
 
 
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 Preste aten\303\247\303\243o no r acima, pois para achar a \303\241rea de uma fun\303\247\303\243o do R^2,\303\251 s\303\263 integrar a fun\303\247\303\243o f(x,y)=1, mas como estamos em coordenadas polares devemos fazer o ajuste que consiste em multiplicar a fun\303\247\303\243o por r, ou seja, ao inv\303\251s de integrar 1, n\303\263s integramos 1*r.
 OBSERVA\303\207\303\203O: A integral que possuir uma fun\303\247\303\243o nos intervalos de integra\303\247\303\243o deve ser a mais interna SEMPRE, j\303\241 que a integral se resolve de dentro para fora!!!
 
 
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 SSNQaUclKnByb3RlY3RlZEc=
 
 
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 JCIrYUVmVEohIio=
 
 
 Com o int de letra min\303\272scula j\303\241 calculamos direto, mas com o de letra mai\303\272scula, s\303\263 mostramos a integral, ent\303\243o damos um evalf para avaliar a integral apresentada. 
 
 
 Portanto, a \303\241rea de R vale Pi u. a..
 
 
 b)
 
 
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 JCIrNV4nUXQnISIq
 
 
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 JCIrT2VYVkAhIio=
 
 
 Primeirose calcula a intregral de x ou y puramente e depois tem que ser dividido pela \303\241rea que vale Pi.para achar o centr\303\263ide.
 
 
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 IiIh
 
 
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 IiIh
 
 
 Logo, o centr\303\263ide de R \303\251 (2,14 , 0).
 
 
 2\302\272Quest\303\243o:
 
 
 a)
 
 
 Pela equa\303\247\303\243o 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 . Ou seja, \303\251 o primeiro octante de todo o elips\303\263ide.
 
 
 
 Sabe-se que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkmbWZyYWNHRiQ2KC1JJW1zdXBHRiQ2JS1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRidGQS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRiM2JC1GPjYkUSI5RidGQUZBLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZQLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkjbW9HRiQ2LVEiK0YnRkEvJSZmZW5jZUdGVS8lKnNlcGFyYXRvckdGVS8lKXN0cmV0Y2h5R0ZVLyUqc3ltbWV0cmljR0ZVLyUobGFyZ2VvcEdGVS8lLm1vdmFibGVsaW1pdHNHRlUvJSdhY2NlbnRHRlUvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0Zkby1GLDYoLUYvNiUtRjI2JVEieUYnRjVGOEY7RkMtRiM2JC1GPjYkUSI0RidGQUZBRktGTkZRRlNGVi1GLzYlLUYyNiVRInpGJ0Y1RjhGO0ZDLUZXNi1RIn5GJ0ZBRlpGZm5GaG5Gam5GXG9GXm9GYG8vRmNvUSYwLjBlbUYnL0Zmb0ZccS1GVzYtUSI9RidGQUZaRmZuRmhuRmpuRlxvRl5vRmBvL0Zjb1EsMC4yNzc3Nzc4ZW1GJy9GZm9GYnEtRj42JEZNRkFGQQ==
 
 
 QyQ+SSJiRzYiLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQtRic2JC1GJzYkSSJ6R0YlL0YwOyIiIS1JJXNxcnRHRig2IywoIiIiRjgqJEkieEdGJSIiIyMhIiIiIioqJEkieUdGJUY7I0Y9IiIlL0ZAO0YzLCQtRjU2IywmRjhGOEY5RjxGOy9GOjtGMyIiJEY4
 
 LCRJI1BpRyUqcHJvdGVjdGVkRyMiIiQiIik=
 
 
 QyQ+SSdEaWdpdHNHNiIiIiYiIiI=
 
 IiIm
 
 
 JSFH QyQtSSZldmFsZkclKnByb3RlY3RlZEc2I0kiYkc2IiIiIg==
 
 JCImInk2ISIl
 
 
 Como ele pede o calculo da integral com quatro casas decimais, eu utilizei digits para delimitar quantas casas eu queria.
 
 
 Teste 2011.1:
 
 
 
 1\302\272Quest\303\243o:
 Seja U a regi\303\243o do espa\303\247o contida na bola de raio 1 centrada na origem e compreendida entre os planos z =1/2 e z =3/4.
 Calcule:
 a) (1.0) O volume de U.
 b) (1.0) O centr\303\263ide de U.
 
 
 
 Como ele fala uma bola, logo \303\251 uma esfera 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 Como z varia entre valores positivos ...z est\303\241 no primeiro octante da esfera...
 
 
 Sabe-se que 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 , ent\303\243o 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, r(z) = + ou - (sqrt(1- LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y+ )).Como j\303\241 temos z, \303\251 o r que varia em fun\303\247\303\243o de z,no entanto s\303\263 vale a ra\303\255z positiva, pois est\303\241 no quadrante positivo.
 
 
 
 Preste aten\303\247\303\243o no r abaixo, pois para achar o volume de uma fun\303\247\303\243o do R^3,\303\251 s\303\263 integrar a fun\303\247\303\243o f(x,y,z)=1, mas como estamos em coordenadas cil\303\255ndricas devemos fazer o ajuste que consiste em multiplicar a fun\303\247\303\243o por r, ou seja, ao inv\303\251s de integrar 1, n\303\263s integramos 1*r.
 
 
 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
 
 LCQqJiMiIiYiI2siIiJJI1BpRyUqcHJvdGVjdGVkR0YnRic=
 
 
 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJWRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI7RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRj1GQA==
 
 JCIlYkMhIiU=
 
 
 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
 
 IiIh
 
 
 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
 
 IiIh
 
 
 J\303\241 era de se imaginar que x fosse igual a zero,pois \303\251 o centro da esfera.
 
 
 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
 
 IiIh
 
 
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 IiIh
 
 
 J\303\241 era de se imaginar que y fosse igual a zero,pois \303\251 o centro da esfera.
 
 
 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYqLUYsNiVRKWludGVncmFsRidGL0YyLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZULUYsNiVRK3NlbWRpdmlkaXJGJ0YvRjJGPS1GLDYlUSVwZWxvRidGL0YyRj0tRiw2JVEndm9sdW1lRidGL0YyRkFGQS1GPjYtUSomY29sb25lcTtGJ0ZBRkNGRkZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGXm8tRiw2JVEkaW50RidGL0YyLUY2NiQtRiM2K0Zgby1GNjYkLUYjNi1GYG8tRjY2JC1GIzYsRistRj42LVEnJnNkb3Q7RidGQUZDRkZGSEZKRkxGTkZQRlJGVS1GLDYlUSJyRidGL0YyLUY+Ni1RIixGJ0ZBRkMvRkdGMUZIRkpGTEZORlBGUi9GVlEsMC4zMzMzMzMzZW1GJ0ZicC1GPjYtUSI9RidGQUZDRkZGSEZKRkxGTkZQRl1vRl9vLUkmbWZyYWNHRiQ2KC1GIzYlLUYsNiVRJXNxcnRGJy9GMEZFRkEtRjY2JC1GIzYkLUkjbW5HRiQ2JFEiN0YnRkFGQUZBRkEtRiM2JC1GXHI2JFEiNEYnRkFGQS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaXIvJSliZXZlbGxlZEdGRS1GPjYtUSMuLkYnRkFGQ0ZGRkhGSkZMRk5GUC9GU1EsMC4yMjIyMjIyZW1GJ0ZVLUZfcTYoLUYjNiVGY3EtRjY2JC1GIzYkLUZccjYkUSIzRidGQUZBRkFGQS1GIzYkLUZccjYkUSIyRidGQUZBRmRyRmdyRmpyRlxzRkFGQUZlcC1GLDYlUSJ0RidGL0YyRltxLUZccjYkUSIwRidGQUZec0ZgdEZfcC1GLDYlUSNQaUYnRmZxRkFGQUZBRmVwRitGW3EtRl9xNigtRlxyNiRGZnJGQUZedEZkckZnckZqckZcc0Zecy1GX3E2KEZbdEZfckZkckZnckZqckZcc0ZBRkEtRj42LVEiO0YnRkFGQ0ZocEZIRkpGTEZORlBGUkZfby8lK2V4ZWN1dGFibGVHRkVGQQ==LCQqJiMiI0QiJDcmIiIiSSNQaUclKnByb3RlY3RlZEdGJ0Yn
 
 
 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
 
 JCIlTTohIiU=
 
 
 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
 
 IyIiJiIiKQ==
 
 
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 JCIlXWkhIiU=
 
 
 J\303\241 era de se imaginar que fosse 0,625, pois \303\251 o meio de z =1/2 e z =3/4.
 
 
 Ent\303\243o, o centr\303\263ide de U \303\251 (0, 0, 0,625).
 
 
 JSFH
 
 
 
 JSFH

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