<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 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 e de sele\303\247\303\243o de estrat\303\251gias para pontos amostrais (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 amostrais ent\303\243o ela converge, com o mesmo limite, para qualquer escolha de divis\303\243o e pontos amostrais. Quando a fun\303\247\303\243o f \303\251 positiva no intervalo 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, este limite \303\251 a \303\241rea entre o gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNictRiw2JVEiZkYnRjRGNy1GOzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZ24tRiM2Ji1GOzYtUSIoRidGPi9GQUY2RkMvRkZGNkZHRklGS0ZNL0ZQUSwwLjE2NjY2NjdlbUYnL0ZTRmFvLUYjNiQtRiw2JVEieEYnRjRGN0Y+LUY7Ni1RIilGJ0Y+Rl5vRkNGX29GR0ZJRktGTUZgby9GU1EsMC4xMTExMTExZW1GJ0Y+RitGPkYrRj5GK0Y+ 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

Se:

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

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHRkBGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkLi4uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== < 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

a largura do subintervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNictRiw2JVEnJiM5MTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y5LyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUC1JJW1zdWJHRiQ2JS1GLDYlUSJ4RicvRjdRJXRydWVGJy9GOlEnaXRhbGljRictRiM2JC1GLDYlUSJpRidGWUZlbkY5LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Y5LUY9Ni1RIj1GJ0Y5RkBGQkZERkZGSEZKRkwvRk9RLDAuMjc3Nzc3OGVtRicvRlJGY28tRiM2KEYrRlMtRj02LVEoJm1pbnVzO0YnRjlGQEZCRkRGRkZIRkpGTC9GT1EsMC4yMjIyMjIyZW1GJy9GUkZbcC1GVDYlRlYtRiM2JkYrLUYjNiZGaW5GZ28tSSNtbkdGJDYkUSIxRidGOUY5RitGOUZcb0YrRjlGK0Y5RitGOQ==

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 integr\303\241vel (por Riemann) 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 Fundamentalde C\303\241lculo, um dos mais importantes resultados de todo o c\303\241lculo, enuncia que integral definida pode ser avaliada usando uma antiderivada do integrando. 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 somas rand\303\264micas diferentes de Riemann para a regra:

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

Ly1JImZHNiI2I0kieEdGJSwoKiQpRiciIiMiIiJGLEYnISIiRitGLA==

no intervalo:

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

NyQiIiEiIiQ=

pareceram convergir para o mesmo valor, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSMyMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y4LUYjNiQtRjU2JFEiMkYnRjhGOC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRjg= ; isto \303\251:

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

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

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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 por Riemann.

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

Se f \303\251 uma fun\303\247\303\243o cont\303\255nua no intervalo fechado LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RIltGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJXRydWVGJy8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGSi1GIzYmLUYsNiVRImFGJy8lJ2l0YWxpY0dGOi9GNlEnaXRhbGljRictRjI2LVEiLEYnRjUvRjlGPS9GPEY6L0Y/Rj1GQEZCRkRGRi9GSVEmMC4wZW1GJy9GTFEsMC4zMzMzMzMzZW1GJy1GLDYlUSJiRidGUkZURjUtRjI2LVEiXUYnRjVGOEY7Rj5GQEZCRkRGRkZIL0ZMUSwwLjExMTExMTFlbUYnRjVGK0Y1, ent\303\243o f \303\251 integr\303\241vel por Riemannem 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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="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">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</Equation> </Text-field>

Esta n\303\243o requer muita discuss\303\243o. Desde que o integrando seja finito 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 d\303\241:

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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>

Suponha que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 um n\303\272mero entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Geometricamente, esta propriedade por intui\303\247\303\243o faz sentido. Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiM2Ji1GOzYtUSIoRidGPi9GQUY2RkMvRkZGNkZHRklGS0ZNL0ZQUSwwLjE2NjY2NjdlbUYnL0ZTRmZuLUYjNiQtRiw2JVEieEYnRjRGN0Y+LUY7Ni1RIilGJ0Y+RllGQ0ZaRkdGSUZLRk1GZW4vRlNRLDAuMTExMTExMWVtRidGPkYrRj5GK0Y+ > 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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 sobre 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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.

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%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"-%%FONTG6%%&TIMESG%'ITALICG"#7-%%TEXTG6%7$$"""!""$""%!""-%)_TYPESETG6#-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"a6"/%'familyGQ0Times~New~Roman6"/%%sizeGQ#126"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"-%%FONTG6%%&TIMESG%'ITALICG"#7-%%VIEWG6$;$""!!""$"#I!""%(DEFAULTG-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-%*AXESTICKSG6$7$$!#5!""$"#S!""""%-%+AXESLABELSG6%-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"x6"/%'familyGQ(DEFAULT6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5-%%ROOTG6'-%)BOUNDS_XG6#$"$S%!""-%)BOUNDS_YG6#$"#]!""-%-BOUNDS_WIDTHG6#$"%SM!""-%.BOUNDS_HEIGHTG6#$"%qO!""-%)CHILDRENG6"

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 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 parti\303\247\303\265es.

Seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA uma parti\303\247\303\243o de 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 em n subintervalos com limites LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHRkBGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkLi4uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== < 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= com a propriedade que:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNipGKy1JJ211bmRlckdGJDYlLUkjbW9HRiQ2L1EkbGltRicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHRlAvRj5RJ2l0YWxpY0YnLUY3Ni1RJyZyYXJyO0YnRj1GQ0ZGRkhGSkZML0ZPRkVGUS9GVFEsMC4yNzc3Nzc4ZW1GJy9GV0Zhby1GLDYlUSgmIzg3MzQ7RidGaG5Gam5GPS8lLGFjY2VudHVuZGVyR0ZFRistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNGVtRicvJSZkZXB0aEdGXXAvJSpsaW5lYnJlYWtHUSVhdXRvRidGKy1GIzYmLUY3Ni1RIihGJ0Y9L0ZERlBGRi9GSUZQRkpGTEZfb0ZRL0ZURlhGVi1GIzYqRistSSttdW5kZXJvdmVyR0YkNictRjc2L1EmJlN1bTtGJ0Y6Rj1GQEZDRkZGXHFGSi9GTUZQRk5GUUZTRlYtRiM2Ji1GLDYlUSJpRidGaG5Gam4tRjc2LVEiPUYnRj1GQ0ZGRkhGSkZMRl9vRlFGYG9GYm8tSSNtbkdGJDYkUSIxRidGPUY9RmVuRlFGZm9GKy1GaW82JkZbcC9GX3BRJDUuMEYnRmFwRmNwRistRiM2KkYrLUYjNictRiw2JVEiZkYnRmhuRmpuLUY3Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj1GQ0ZGRkhGSkZMRl9vRlFGUy9GV0ZVLUYjNiZGaHAtRiM2JkYrLUklbXN1YkdGJDYlLUYsNiVRInVGJ0ZobkZqbi1GIzYkRmlxRj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRj0tRjc2LVEiKUYnRj1GW3FGRkZccUZKRkxGX29GUUZdcS9GV1EsMC4xMTExMTExZW1GJ0Y9RitGPS1GNzYtUTEmSW52aXNpYmxlVGltZXM7RidGPUZDRkZGSEZKRkxGX29GUUZTRmFzLUYsNiVRJyYjOTE2O0YnL0ZpbkZFRj1GZnRGZnNGK0Y9RitGPUZhdEY9RitGPUZcci1GIzYqRistSShtc3Vic3VwR0YkNictRjc2L1EmJmludDtGJ0Y6Rj1GQEZDRkZGSEZKRkxGX29GUUZTRmFzLUYsNiVRImFGJ0ZobkZqbi1GLDYlUSJjRidGaG5Gam4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnRl50RistRiM2J0Zbc0Zecy1GIzYmRmhwLUYjNiQtRiw2JVEieEYnRmhuRmpuRj1GYXRGPUYrRj1GKy1GaW82JkZbcC9GX3BRJjAuM2VtRidGYXBGY3AtRiw2JVEjZHhGJ0ZobkZqbkY9RitGPUYrRj0=

e seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA uma divis\303\243o de 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 em n subintervalos com limites LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHRkBGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkLi4uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== < 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= com a propriedade que:

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

Os pontos 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 , 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkLi4uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== 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 formam uma divis\303\243o 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRIm5GJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= subintervalos com larguras 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPi1GOzYtUSIsRidGPkZAL0ZERjZGRUZHRklGS0ZNL0ZQUSYwLjBlbUYnL0ZTUSwwLjMzMzMzMzNlbUYnLUZVNiRRIjJGJ0Y+RlgtRiw2JVEkLi4uRidGNEY3RlgtRiw2JVEibkYnRjRGN0Y+. Ent\303\243o, para cada inteiro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=,

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e a linearidade 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 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 existe, isto pode ser identificado e escrito como uma integral definida:

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

Embora esta prova 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 positiva. Encontre a \303\241rea da regi\303\243o no primeiro quadrante limitada pelo gr\303\241fico da fun\303\247\303\243o definida por partes que tem a regra:

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

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNihGKy1JJW1zdWJHRiQ2JS1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIyRicvRj1RJ25vcm1hbEYnRkUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRkUvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlAvJSlzdHJldGNoeUdGUC8lKnN5bW1ldHJpY0dGUC8lKGxhcmdlb3BHRlAvJS5tb3ZhYmxlbGltaXRzR0ZQLyUnYWNjZW50R0ZQLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGaW4tRiM2Ji1GSzYtUSIoRidGRS9GT0Y7RlEvRlRGO0ZVRldGWUZlbi9GaG5RLDAuMTY2NjY2N2VtRicvRltvRmRvLUYjNiQtRiw2JVEieEYnRjlGPEZFLUZLNi1RIilGJ0ZFRmFvRlFGYm9GVUZXRllGZW5GY28vRltvUSwwLjExMTExMTFlbUYnRkVGK0ZFLUZLNi1RIj1GJ0ZFRk5GUUZTRlVGV0ZZRmVuL0ZoblEsMC4yNzc3Nzc4ZW1GJy9GW29GZHAtSSZtZnJhY0dGJDYoLUYjNiZGKy1GIzYnRistRiM2Ji1GQjYkUSIzRidGRS1GSzYtUTEmSW52aXNpYmxlVGltZXM7RidGRUZORlFGU0ZVRldGWUZlbkZnbkZqbi1GLDYlUSJhRidGOUY8RkUtRks2LVEoJm1pbnVzO0YnRkVGTkZRRlNGVUZXRllGZW4vRmhuUSwwLjIyMjIyMjJlbUYnL0Zbb0ZcckZob0ZFRitGRUY/LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zjci8lKWJldmVsbGVkR0ZQRitGRUYrRkU=

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:

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

LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmMkkieEc2IkkiYUdGKCokKSwmKigiIiMiIiJGKUYvRidGL0YvKiQpRidGLkYvISIiI0YvRi5GLzFGJywkKiYiIiRGL0YpRi9GLywmKiYjRjdGLkYvRilGL0YvKiZGM0YvRidGL0Yy

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

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

Ly1JImZHNiI2I0kieEdGJS1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JjMxIiIhRicyRidJImFHRiUqJCksJiooIiIjIiIiRjBGNkYnRjZGNiokKUYnRjVGNiEiIiNGNkY1RjYzMUYwRicxRicsJComIiIkRjZGMEY2RjYsJiomI0ZARjVGNkYwRjZGNiomRjpGNkYnRjZGOQ==

Por causa do par\303\242metro 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=.

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNQM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

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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 e um tr\303\242ngulo, como mostrado na Figura 24.4.

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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 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RIltGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJXRydWVGJy8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGSi1GIzYnLUYsNiVRImFGJy8lJ2l0YWxpY0dGOi9GNlEnaXRhbGljRictRjI2LVEiLEYnRjUvRjlGPS9GPEY6L0Y/Rj1GQEZCRkRGRi9GSVEmMC4wZW1GJy9GTFEsMC4zMzMzMzMzZW1GJy1GIzYmLUkjbW5HRiQ2JFEiM0YnRjUtRjI2LVExJkludmlzaWJsZVRpbWVzO0YnRjVGWUY7RmVuRkBGQkZERkZGZm4vRkxGZ25GT0Y1RitGNS1GMjYtUSJdRidGNUY4RjtGPkZARkJGREZGRkgvRkxRLDAuMTExMTExMWVtRidGNUYrRjU=. Logo, a \303\241rea sob a fun\303\247\303\243o pode ser avaliada como:

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

= 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

= 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

= 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

O Exemplo 24.2 demonstra a integral definida de uma fun\303\247\303\243o cont\303\255nua definida por partes, um conceito tornado preciso 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 consecutivos 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, uma descontinuidade de salto. A fun\303\247\303\243o n\303\243o pode se tornar ilimitada nem ter oscila\303\247\303\265es infinitas em nenhum destes pontos. Logo, comportamentos tais como de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjlRJ2l0YWxpY0YnRjgvJS5saW5ldGhpY2tuZXNzR0Y3LyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkkvJSliZXZlbGxlZEdRJmZhbHNlRidGK0Y4 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= est\303\243o exclu\303\255dos, 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 integr\303\241vel por Riemann em 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.

Uma conseq\303\274\303\252ncia deste teorema \303\251 surpreendentemente significante. O valor de um integrando integr\303\241vel por Riemann pode ser mudado em um n\303\272mero finito de pontos no dom\303\255nio de integra\303\247\303\243o, desde que estes novos valores sejam todos finitos, o valor da 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 integr\303\241vel por Riemann 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 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, em um computador \303\251 racional, um gr\303\241fico gerado por computador apareceria com a reta horizontal LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=.

Seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA a divis\303\243o de 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 em n subintervalos com 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. Ent\303\243o 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvRjlRJ2l0YWxpY0YnRjgvJS5saW5ldGhpY2tuZXNzR0Y3LyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkkvJSliZXZlbGxlZEdRJmZhbHNlRidGK0Y4 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 inferiores convergem para 0. Como 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 integr\303\241vel por Riemann.

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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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KUYrLUYjNitGKy1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNjBRJiZpbnQ7RicvJSVib2xkR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEmaW5lcnRGJy8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlYtRiw2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnL0ZBUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GLDYmUSJhRidGZm5GaW5GW28vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1GIzYoLUYsNiZRImZGJ0ZmbkZpbkZbby1GNzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkZGSEZKRkxGTkZQRlJGVEZXLUYjNictRjc2LlEiKEYnRjpGQC9GR0ZobkZIL0ZLRmhuRkxGTkZQRlIvRlVRLDAuMTY2NjY2N2VtRicvRlhGZ3AtRiM2JS1GLDYmUSJ4RidGZm5GaW5GW28vRkFGXW9GW28tRjc2LlEiKUYnRjpGQEZkcEZIRmVwRkxGTkZQRlJGZnAvRlhRLDAuMTExMTExMWVtRidGXnFGW29GK0ZecUZbb0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmlxLyUqbGluZWJyZWFrR1ElYXV0b0YnLUYsNiZRI2R4RidGZm5GaW5GW29GXnFGW28tRjc2LlEiPUYnRjpGQEZGRkhGSkZMRk5GUEZSL0ZVUSwwLjI3Nzc3NzhlbUYnL0ZYRmlyLUYjNictRjc2LlEqJnVtaW51czA7RidGOkZARkZGSEZKRkxGTkZQRlJGVEZXLUYjNitGKy1GNDYnRjZGXm9GWUZhb0Zkb0YrRmdvRitGZHFGYnJGXnFGW29GK0ZecUZbb0YrRl5xRltvRitGXnFGW28=</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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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrRistSSNtb0dGJDYtUSI9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktRiM2KkYrLUknbXVuZGVyR0YkNiUtRjI2L1EkbGltRicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnRjUvSSttc2VtYW50aWNzR0YkUSZpbmVydEYnRjhGO0Y9Rj9GQS9GRFEldHJ1ZUYnRkUvRkhRJjAuMGVtRicvRktRLDAuMTY2NjY2N2VtRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHRmVuL0Y2USdpdGFsaWNGJy1GMjYtUScmcmFycjtGJ0Y1RjhGO0Y9Rj9GQUZDRkVGR0ZKLUYsNiVRKCYjODczNDtGJ0Zfb0Zhb0Y1LyUsYWNjZW50dW5kZXJHRjpGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC40ZW1GJy8lJmRlcHRoR0ZgcC8lKmxpbmVicmVha0dRJWF1dG9GJ0YrLUYjNiYtRjI2LVEiKEYnRjUvRjlGZW5GOy9GPkZlbkY/RkFGQ0ZFL0ZIRmluRmhuLUYjNipGKy1JK211bmRlcm92ZXJHRiQ2Jy1GMjYvUSYmU3VtO0YnRlRGNUZXRjhGO0ZfcUY/L0ZCRmVuRlpGRUZmbkZobi1GIzYmLUYsNiVRImlGJ0Zfb0Zhb0YxLUkjbW5HRiQ2JFEiMUYnRjVGNUZcb0ZFRmlvRistRlxwNiZGXnAvRmJwUSQ1LjBGJ0ZkcEZmcEYrLUYjNiZGW3EtRiM2KEYrLUYjNictRiw2JVEickYnRl9vRmFvLUYyNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y1RjhGO0Y9Rj9GQUZDRkVGZm4vRktGZ24tRiM2Jy1GLDYlUSJmRidGX29GYW8tRjI2LVEwJkFwcGx5RnVuY3Rpb247RidGNUY4RjtGPUY/RkFGQ0ZFRmZuRmNzLUYjNiZGW3EtRiM2JkYrLUklbXN1YkdGJDYlLUYsNiVRInhGJ0Zfb0Zhby1GIzYkRlxyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRjUtRjI2LVEiKUYnRjVGXnFGO0ZfcUY/RkFGQ0ZFRmBxL0ZLUSwwLjExMTExMTFlbUYnRjVGK0Y1RitGNS1GMjYtUSIrRidGNUY4RjtGPUY/RkFGQ0ZFL0ZIUSwwLjIyMjIyMjJlbUYnL0ZLRmR1LUYjNictRiw2JVEic0YnRl9vRmFvRmBzLUYjNictRiw2JVEiZ0YnRl9vRmFvRmlzRlx0RitGNUYrRjVGK0Y1Rlt1RjVGK0Y1Rlt1RjVGK0Y1RitGNUYrRjU=

= 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

= 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

= 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

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

No Exemplo 24.1 foi determinado que:

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). A segunda e a terceira integrais do lado direito podem ser determinadas pela geometria, como resumimos na Tabela 24.1.

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_WIDTHG6#$"%![$!""-%.BOUNDS_HEIGHTG6#$"%g6!""-%)CHILDRENG6"

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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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GNTYkUSIyRidGOEY4LyUubGluZXRoaWNrbmVzc0dGNy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZELyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== (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