<Text-field style="Heading 1" layout="Heading 1">Introdu\303\247\303\243o</Text-field>

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">A Integral Definida</Text-field>

A discuss\303\243o de somas de Riemann no cap\303\255tulo anterior considera um n\303\272mero de diferentes escolhas de divis\303\243o do intervalo (interval) 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 e de sele\303\247\303\243o de estrat\303\251gias para pontos amostras (direita, esquerda, ponto m\303\251dio, rand\303\264mico, superior, inferior) dentro de cada subintervalo. Os exemplos sugerem que, no limite quando a largura da divis\303\243o decresce tendendo a zero, estas escolhas n\303\243o s\303\243o importantes. Isto \303\251, se o limite existe para uma escolha de divis\303\265es e de pontos amostras ent\303\243o ela converge, com o mesmo limite, para qualquer escolha de divis\303\243o e pontos amostras. Quando a fun\303\247\303\243o (function) f \303\251 positiva (positive) no intervalo interval 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, este limite \303\251 a \303\241rea entre o gr\303\241fico (graph) de 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 e o eixo-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= sobre o intervalo 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. De modo mais geral, estes limites de somas de Riemann s\303\243o chamados integrais definidas, como por Defini\303\247\303\243o 24.1.

Defini\303\247\303\243o 24.1: A Integral Definida

f \303\251 uma fun\303\247\303\243o definida em um intervalo fechado 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 .

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 a divis\303\243o de 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 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= subintervalos com limites

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkLi4uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== < 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

a largura do subintervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 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

a largura da divis\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 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 \303\251 um ponto no subintervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, i.e., 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUklbXN1YkdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRictRiM2JC1GLDYlUSJpRidGUkZVRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRjVGK0Y1

ent\303\243o

A integral definida da fun\303\247\303\243o f sobre o intervalo 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 \303\251

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

desde que o limite exista. Quando esta integral definida existe diz-se que f \303\251 (Riemann) integr\303\241vel em 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.

Neste ponto as \303\272nicas op\303\247\303\265es para avaliar uma integral definida s\303\243o i) usar a defini\303\247\303\243o em termos de limites e ii) reconhecer a integral definida como a \303\241rea conhecida de uma regi\303\243o geom\303\251trica. O (segundo) Teorema Fundamental de C\303\241lculo, um dos mais importantes resultados de todo o c\303\241lculo (diferencial integral), enuncia que integral definida pode ser avaliada usando uma antiderivada do integrando (integrand). Os cap\303\255tulos que concluem esta unidade cont\303\251m este resultado e outras t\303\251cnicas para a avalia\303\247\303\243o de integrais definidas.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 24.1</Text-field>

No cap\303\255tulo Somas de Riemann, as somas de Riemann esquerda, direita e duas diferente somas de Riemann para a regra

f1 := x^2 - x + 2: f(x) = f1;

no intervalo

a1 := 0: b1 := 3: [a1,b1];

todas pareceram convergir (converge) para o mesmo valor, a saber, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSMyMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y4LUYjNiQtRjU2JFEiMkYnRjhGOC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRjg= ; isto \303\251,

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

A Figure 24.1 cont\303\251m o gr\303\241fico do integrando

p1 := plot(f1,x=a1..b1,color=black, thickness=3): p2 := plot(f1,x=a1..b1,filled=true,color=cyan): display([p1,p2], tickmarks=[3,4]);

Figura 24.1 O integrando e a \303\241rea do Exemplo 24.1

A regi\303\243o sombreada \303\251 a \303\241rea computada pela integral definida considerada.

O Teorema 24.1, enunciou sem provar, define uma grande classe de fun\303\247\303\265es integr\303\241veis segundo de Riemann ou Riemann integr\303\241veis (Riemann integrable) .

Teorema 24.1: Exist\303\252ncia da Integral de Riemann

Se f \303\251 uma fun\303\247\303\243o cont\303\255nua (continuous) no intervalo fechad 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, ent\303\243o f \303\251 Riemann integr\303\241vel em 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.

As classes de fun\303\247\303\265es integr\303\241veis ser\303\243o expandidas mais tarde neste cap\303\255tulo. Por\303\251m, primeiro, vamos desenvolver algumas propriedades para integrais definidas.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Propriedade 1: <Equation executable="false" style="2D Math_113" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNitGKy1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNjBRJiZpbnQ7RicvJSVib2xkR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEmaW5lcnRGJy8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlYtRiw2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0ZBUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJ0ZZLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjJGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRistRiM2KC1GLDYmUSJmRidGZm5GaW5GW28tRjc2LVEwJkFwcGx5RnVuY3Rpb247RidGQEZGRkhGSkZMRk5GUEZSRlRGVy1GIzYnLUY3Ni5RIihGJ0Y6RkAvRkdGaG5GSC9GS0ZobkZMRk5GUEZSL0ZVUSwwLjE2NjY2NjdlbUYnL0ZYRmRwLUYjNiUtRiw2JlEieEYnRmZuRmluRltvL0ZBRl1vRltvLUY3Ni5RIilGJ0Y6RkBGYXBGSEZicEZMRk5GUEZSRmNwL0ZYUSwwLjExMTExMTFlbUYnRltxRltvRitGW3FGW29GKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0ZmcS8lKmxpbmVicmVha0dRJWF1dG9GJy1GLDYmUSNkeEYnRmZuRmluRltvRltxRltvLUY3Ni5RIj1GJ0Y6RkBGRkZIRkpGTEZORlBGUi9GVVEsMC4yNzc3Nzc4ZW1GJy9GWEZmci1JI21uR0YkNiVRIjBGJ0ZbcUZbb0ZbcUZbb0YrRltxRltv">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</Equation> </Text-field>

Esta n\303\243o requer muita discuss\303\243o. Desde que o integrando seja finito (finite) em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4=, a regi\303\243o ser\303\241 um "ret\303\242ngulo" com altura 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 e "largura" 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. A \303\241rea deste ret\303\242ngulo \303\251 0. Enunciada em termos de integrais definidas, esta propriedade l\303\252

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

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Propriedade 2: <Equation executable="false" style="2D Math_114" input-equation="" display="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">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</Equation> </Text-field>

Supondo que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 um n\303\272mero entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Geometricamente, esta propriedade por intui\303\247\303\243o faz sentido. Se 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 > 0 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInhGJ0Y0RjdGPkYrRj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYsNiVRImJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, ent\303\243o a \303\241rea sob o gr\303\241fico de 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 sobre 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 \303\251 a soma das \303\241reas sob o gr\303\241fico de 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 e sobre 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.

A Figura 24.2 ilustra a conex\303\243o entre a \303\241rea e as integrais na Propriedade 2.

P1 := plot( f1, x=0..2, color=cyan, filled=true ): P2 := plot( f1, x=2..3, color=pink, filled=true ): P3 := textplot({[.1,.4,a],[2,.4,c],[3,.4,b]}, font=[TIMES,ITALIC,12]): display(P1,P2,P3, tickmarks=[[-1,4],4]);

Figura 24.2

Interpreta\303\247\303\243o geom\303\251trica da Propriedade 2: 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

Se o integrando tem alguns valores negativos (negative) em 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, ent\303\243o n\303\243o \303\251 poss\303\255vel usar a \303\241rea para justificar esta propriedade. Em vez disto \303\251 necess\303\241rio olhar para as divis\303\265es.

Seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA uma divis\303\243o de 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 em n subintervalos com limites (endpoints) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkLi4uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== < 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= com a propriedade que

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 , para 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. Ent\303\243o, para cada inteiro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=,

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e a linearidade (linearity) dos limites fornecem

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

= 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

= 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

Agora, como o limite original de somas de Riemann sobre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RIltGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJXRydWVGJy8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGSi1GIzYmLUYsNiVRImFGJy8lJ2l0YWxpY0dGOi9GNlEnaXRhbGljRictRjI2LVEiLEYnRjUvRjlGPS9GPEY6L0Y/Rj1GQEZCRkRGRi9GSVEmMC4wZW1GJy9GTFEsMC4zMzMzMzMzZW1GJy1GLDYlUSJiRidGUkZURjUtRjI2LVEiXUYnRjVGOEY7Rj5GQEZCRkRGRkZIL0ZMUSwwLjExMTExMTFlbUYnRjVGK0Y1 existe, isto pode ser identificado e escrito como uma integral definida, isto \303\251,

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

Embora esta prova (proof) seja v\303\241lida somente quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImNGJ0Y0RjdGPkYrRj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYsNiVRImJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, o resultado \303\251 verdadeiro para qualquer valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para o qual todas as tr\303\252s integrais existam.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 24.2: Integrando Cont\303\255nuo por Partes </Text-field>

Seja a uma constante (constant) positiva. Encontre a \303\241rea da regi\303\243o no primeiro quadrante (quadrant) limitada pelo gr\303\241fico da fun\303\247\303\243o definida por partes (piecewise-defined) que tem a regra

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

quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVEiPEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYsNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, e a regra

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

quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInhGJ0Y0RjdGPkYrRj4=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.

A seguinte sintaxe \303\251 a mais simples para se entrar com esta fun\303\247\303\243o no Maple

f2 := piecewise(x<a,sqrt(2*a*x-x^2), x<=3*a, (3*a-x)/2);

A seguinte tentativa para entrar com a fun\303\247\303\243o exatamente no dom\303\255nio (domain) 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 resulta em uma sa\303\255da do Maple que \303\251 excessivamente dif\303\255cil de se ler.

f(x) = piecewise( x>=0 and x<a, sqrt( 2*a*x-x^2 ), x>=a and x<= 3*a, (3*a-x)/2 );

Por causa do par\303\242metro (parameter) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, n\303\243o \303\251 poss\303\255vel para o Maple fazer o gr\303\241fico da regi\303\243o limitada pelo eixo-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e o gr\303\241fico desta fun\303\247\303\243o 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. Entretanto, pode se fazer o gr\303\241fico para valores espec\303\255ficos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Por exemplo, Figura 24.3 mostra a regi\303\243o quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GPkYrRj4=.

P3 := plot( eval(f2,a=2), x=0..6 ): P3;

Figura 24.3 A fun\303\247\303\243o por partes do Exemplo 24.2

Esta regi\303\243o \303\251 claramente a uni\303\243o de um quarto de c\303\255rculo (circle) e um tr\303\242ngulo, como mostrado na Figura 24.4.

P4 := plot(eval(f2,a=2), x=0..2, color=pink, filled=true): P5 := plot(eval(f2,a=2), x=2..6, color=cyan, filled=true): display( P3,P4,P5 );

Figura 24.4

Exemplo 24.2: A \303\241rea sob LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== \303\251 a uni\303\243o de um quarto de c\303\255rculo e um tri\303\242ngulo

A \303\241rea da regi\303\243o rosa \303\251 um quarto de c\303\255rculo com raio (radius) 2. A \303\241rea da regi\303\243o azul \303\251 um tr\303\242ngulo de base 4 e altura 2. Portanto, pela Propriedade 2,

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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiK0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFEiNEYnRjdGN0YrRjc=

De modo mais geral, a regi\303\243o consiste da uni\303\243o de um quarto de c\303\255rculo com raio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= sobre 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 e um tri\303\242ngulo com base LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= e altura LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= sobre 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. Logo, a \303\241rea sob a fun\303\247\303\243o pode ser computada como

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

= 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

= 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

= 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

O Exemplo 24.2 demonstra o integral definida de uma fun\303\247\303\243o cont\303\255nua definida por partes , um conceito preciso \303\251 feito na Defini\303\247\303\243o 24.2.

Defini\303\247\303\243o 24.2: Cont\303\255nua por Partes

Uma fun\303\247\303\243o cont\303\255nua por partes \303\251 aquela que \303\251 cont\303\255nua em cada um dos intervalos de uma cole\303\247\303\243o de intervalos cont\303\255guos que n\303\243o se sobrep\303\265em

Portanto, nestes pontos onde intervalos se encontram, a fun\303\247\303\243o pode ter, no pior dos casos, um salto de descontinuidade (jump discontinuity). A fun\303\247\303\243o n\303\243o pode se tornar ilimitada (unbounded) nem ter oscila\303\247\303\265es infinitas em nenhum destes pontos. Logo, comportamentos tais como de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjlRJ2l0YWxpY0YnRjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkovJSliZXZlbGxlZEdRJmZhbHNlRidGK0Y4 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= est\303\241 exclu\303\255do, e tamb\303\251m comportamentos como o de 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 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=.

Em termos da Defini\303\247\303\243o 24.2, Podemos enunciar o Teorema 24.2.

Teorema 24.2: Integrabilidade de Riemann de Fun\303\247\303\265es Cont\303\255nuas por Partes

Se f \303\251 uma fun\303\247\303\243o cont\303\255nua por partes no intervalo 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, ent\303\243o f \303\251 Riemann integr\303\241vel em 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.

Uma conseq\303\274\303\252ncia deste teorema \303\251 surpreendentemente significante. O valor de um integrando integral de Riemann pode ser mudado em um n\303\272mero finito de pontos no alcance (range) da integra\303\247\303\243o, desde que estes novos valores sejam todos finito, o valor da integral (integral) n\303\243o modificar\303\241.

<Text-field style="Heading 3" layout="Heading 3">Esquema da Prova do Teorema 24.2</Text-field>

Dividir o intervalo 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 em um n\303\272mero finito de intervalos n\303\243o superpostos, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkLi4uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRImtGJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA , de modo que em cada um deles f seja cont\303\255nua. Ent\303\243o, pelo Teorema 1, f \303\251 integr\303\241vel em cada um dos intervalos menores. A Propriedade 2 nos diz que a soma destas integrais definidas \303\251 a integral definida sobre 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. Como esta integral definida existe, f deve ser Riemann integr\303\241vel em todo intervalo, 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

<Text-field superscript="false" placeholder="false" executable="false" selection-placeholder="false" italic="false" size="16" bold="false" subscript="false" family="Arial" opaque="false" underline="false" background="[255,255,255]" readonly="false" foreground="[0,0,0]" alignment="left" firstindent="0" spacebelow="0" leftmargin="0" linespacing="0.0" initial="0" linebreak="space" rightmargin="0" bulletsuffix="" spaceabove="0" bullet="none" pagebreak-before="false"><Font superscript="false" placeholder="false" executable="false" selection-placeholder="false" italic="false" size="16" bold="true" subscript="false" family="Arial" opaque="false" underline="false" background="[0,0,0]" encoding="UTF-8" readonly="false" foreground="[0,0,0]">Exemplo 24.3: Uma Fun\303\247\303\243o N\303\243o Integr\303\241vel </Text-field>

Considere a fun\303\247\303\243o

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

Lembre-se que entre dois n\303\272meros racionais (rational) quaisquer, existe pelo menos um n\303\272mero irracional e que entre dois n\303\272meros irracionais quaisquer existe pelo menos um n\303\272mero racional. Da forma como os n\303\272meros racionais e irracionais est\303\243o intercalados, n\303\243o \303\251 poss\303\255vel criar um gr\303\241fico instrutivo desta fun\303\247\303\243o. Pelo fato de que cada n\303\272mero, represent\303\241vel com ponto flutuante (floating-point) , em um computador \303\251 racional, um gr\303\241fico gerado por computador apareceria com a linha horizontal LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=.

Seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA a divis\303\243o de 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 em n subintervalos com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUklbXN1YkdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLDYlUSJpRidGN0Y6L0Y7USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRkIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk0vJSlzdHJldGNoeUdGTS8lKnN5bW1ldHJpY0dGTS8lKGxhcmdlb3BHRk0vJS5tb3ZhYmxlbGltaXRzR0ZNLyUnYWNjZW50R0ZNLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZm4tSSZtZnJhY0dGJDYoRj0tRiM2JC1GLDYlUSJuRidGN0Y6RkIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZvLyUpYmV2ZWxsZWRHRk1GK0ZCRitGQg== . Ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNictRiw2JVEnJiM5MTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y5LyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUC1JJW1zdWJHRiQ2JS1GLDYlUSJ4RicvRjdRJXRydWVGJy9GOlEnaXRhbGljRictRiM2JC1GLDYlUSJpRidGWUZlbkY5LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Y5LUY9Ni1RIj1GJ0Y5RkBGQkZERkZGSEZKRkwvRk9RLDAuMjc3Nzc3OGVtRicvRlJGY28tRiM2JkYzRjxGVkY5RitGOUYrRjk= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvRjlRJ2l0YWxpY0YnRjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkovJSliZXZlbGxlZEdRJmZhbHNlRidGK0Y4 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . Como cada subintervalo cont\303\251m um n\303\272mero racional, o maior valor de f no subintervalo \303\251 1. Portanto a soma de Riemann superior em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA \303\251

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 = 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

Da mesma forma, o menor valor em cada subintervalo \303\251 0 e a soma de Riemann inferior de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA \303\251

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 = 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

A seq\303\274\303\252ncia das somas de Riemann superiores convergem para 1 e a seq\303\274\303\252ncia das somas de Riemann infariores convergem para 0. Porque as somas de Riemann superiores e inferiores para a mesma fun\303\247\303\243o e intervalo n\303\243o convergem para o mesmo valor, esta fun\303\247\303\243o n\303\243o \303\251 Riemann integr\303\241vel.

A fun\303\247\303\243o no Exemplo 24.3 \303\251 diferente de qualquer outra fun\303\247\303\243o encontrada neste curso. Ela est\303\241 inclu\303\255da aqui somente para mostrar que existem, de fato, algumas fun\303\247\303\265es que n\303\243o s\303\243o integr\303\241veis. N\303\243o se encontrar\303\241, normalmente, outra fun\303\247\303\243o como esta nas aplica\303\247\303\265es t\303\255picas de c\303\241lculo elementar.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Propriedade 3: <Equation executable="false" style="2D Math_115" input-equation="" display="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">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</Equation> </Text-field>

Pela Propriedade 2 sabe-se que

Pela Propriedade 1 o lado esquerdo desta equa\303\247\303\243o \303\251 0, assim

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

segue que

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Propriedade 4: <Equation executable="false" style="2D Math_117" input-equation="" display="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">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</Equation> </Text-field>

A Propriedade 4 enuncia a propriedade de linearidade para integrais definidas. A linearidade da integral definida \303\251 uma conseq\303\274\303\252ncia direta da linearidade de limites e somas, como a seguinte prova mostra.

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<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 24.4</Text-field>

No Exemplo 24.1 foi determinadoque

Usar as propriedades de integrais definidas para determinar 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.

A linearidade da integral definida, Propriedade 4, nos permite escrever

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

= 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

A integral definida \303\240 esquerda \303\251 conhecida (do Exemplo 24.1). As segunda e terceira integrais do lado direito podem ser determinadas pela geometria, como resumimos na Tabela 24.1.

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

JSFHp3 := plot(x, x=0..3, color=black): p4 := plot(x, x=0..3, color= pink, filled=true): display([p3,p4]);

JSFHp5 := plot(1, x=0..3, color=black): p6 := plot(1, x=0..3, color=cyan, filled=true): display([p5,p6], scaling=constrained);

Figura 24.5 Gr\303\241fico do integrando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyRjk=

Figura 24.6 Gr\303\241fico do integrando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=

A \303\241rea do tri\303\242ngulo reto com base 3 e altura 3 \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSI5RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GNTYkUSIyRidGOEY4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZFLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== , tal que

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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GNTYkUSIyRidGOEY4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZFLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== (3) (3) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSI5RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GNTYkUSIyRidGOEY4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZFLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA==

A \303\241rea do ret\303\242ngulo com base 3 e altura 1 \303\251 3, tal que

Tabela 24.1 Interpreta\303\247\303\243o geom\303\251trica de duas das integrais definidas no Exemplo 24.4

Portanto, podemos escrever

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

= 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

= 9