Calcule as integrais definidas (1) $\int_{-3}^{3}\left|x^{4}-9\right| d x$
Passo 1
Beleza, vamos resolver essa integral definida:
. A matéria envolvida aqui é o cálculo de integrais definidas e o funcionamento do valor absoluto dentro da integral. Para resolver, vamos precisar entender onde a expressão
é positiva ou negativa, assim podemos remover o valor absoluto de forma correta e calcular a área total.
Passo 2
Primeiro, vamos descobrir onde
é igual a zero, porque é aí que a função muda de positiva para negativa ou vice-versa. Então, resolvemos a equação:
Portanto, os pontos onde a função cruza o eixo x são aproximadamente
e
.
Passo 3
Agora, vamos analisar os intervalos determinados por esses pontos para ver o sinal de
:
-
Para
: Vamos testar . -
Para
: Vamos testar . -
Para
: Vamos testar .
Passo 4
Com isso, sabemos que a função
é positiva nos intervalos
e
, e negativa no intervalo
. Isso significa que o valor absoluto vai inverter o sinal apenas no meio. Então, podemos reescrever a integral dividindo-a nesses intervalos:
Passo 5
Vamos calcular cada uma dessas integrais separadamente.
Primeiro, a integral de
:
E a integral de
:
Passo 6
Agora, vamos calcular cada parte:
-
: -
: -
:
Passo 7
Vamos substituir os limites de integração em cada parte:
-
Para
: -
Para
: -
Para
:
Passo 8
Calculando cada termo:
-
Então,
-
Então,
-
Então,
Resposta
Somando todas as partes:
Portanto, o valor da integral definida é aproximadamente 91,88.
Exercícios relacionados
Calcule as integrais definidas
(i)
(i)
Calcule as integrais definidas
©
©
<html><head></head><body><div>Calcule a área das regiões limitadas pelas curvas dadas:<br>
(a) <span class="math-inline ">
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(a) <span class="math-inline ">
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Nos itens abaixo encontre a curva característica e uma forma canônica para cada equação