Integrais Triplas em Coordenadas Esféricas

 Integrais Triplas em Coordenadas Esféricas

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Introdução:

Nesta publicação, vamos explorar um tópico fascinante em cálculo: integrais triplas em coordenadas esféricas. Essa técnica é uma ferramenta poderosa para calcular volumes, embora não seja essa uma interpretação para a integral tripla, e outras grandezas em três dimensões.

O que são coordenadas esféricas?

Antes de mergulharmos nas integrais triplas em coordenadas esféricas, vamos revisar o que são coordenadas esféricas. Em vez de usar os sistemas cartesianos tradicionais

$(�,�,�)$$�$, o ângulo azimutal $�$, e o ângulo de inclinação $�$.

Transformação de Coordenadas:

Agora, vamos entender como fazer a transformação de coordenadas de cartesianas para esféricas e vice-versa. A relação entre coordenadas cartesianas e esféricas é dada por:

$$\begin{array}{rl}{\displaystyle \mathrm{<span\; style="font-family:\; inherit;">�</span>}}& {\displaystyle <span\; style="font-family:\; inherit;">=</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">sin</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>\mathrm{<span\; style="font-family:\; inherit;">cos</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>}\\ {\displaystyle \mathrm{<span\; style="font-family:\; inherit;">�</span>}}& {\displaystyle <span\; style="font-family:\; inherit;">=</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">sin</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>\mathrm{<span\; style="font-family:\; inherit;">sin</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>}\\ {\displaystyle \mathrm{<span\; style="font-family:\; inherit;">�</span>}}& {\displaystyle <span\; style="font-family:\; inherit;">=</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">cos</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>}\end{array}$$

E podemos expressar $�$, $�$, e $�$ em termos de $�$, $�$, e $�$ usando:

$$\begin{array}{rl}{\displaystyle \mathrm{<span\; style="font-family:\; inherit;">�</span>}}& {\displaystyle <span\; style="font-family:\; inherit;">=</span>\sqrt{{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}^{\mathrm{<span\; style="font-family:\; inherit;">2</span>}}<span\; style="font-family:\; inherit;">+</span>{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}^{\mathrm{<span\; style="font-family:\; inherit;">2</span>}}<span\; style="font-family:\; inherit;">+</span>{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}^{\mathrm{<span\; style="font-family:\; inherit;">2</span>}}}}\\ {\displaystyle \mathrm{<span\; style="font-family:\; inherit;">�</span>}}& {\displaystyle <span\; style="font-family:\; inherit;">=</span>\mathrm{<span\; style="font-family:\; inherit;">arccos</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\frac{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}{\sqrt{{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}^{\mathrm{<span\; style="font-family:\; inherit;">2</span>}}<span\; style="font-family:\; inherit;">+</span>{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}^{\mathrm{<span\; style="font-family:\; inherit;">2</span>}}<span\; style="font-family:\; inherit;">+</span>{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}^{\mathrm{<span\; style="font-family:\; inherit;">2</span>}}}}<span\; style="font-family:\; inherit;">)</span>}\\ {\displaystyle \mathrm{<span\; style="font-family:\; inherit;">�</span>}}& {\displaystyle <span\; style="font-family:\; inherit;">=</span>\mathrm{<span\; style="font-family:\; inherit;">arctan</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\frac{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}<span\; style="font-family:\; inherit;">)</span>}\end{array}$$

Formulação da Integral Tripla:

Compreendida a transformação de coordenadas, vamos agora ver como expressar uma integral tripla em coordenadas esféricas. A forma geral da integral tripla é:

$$\underset{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}{<span\; style="font-family:\; inherit;">\iiint </span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">,</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">,</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}$$

Podemos reescrever isso em termos de $�$, $�$, e $�$ usando a transformação que acabamos de ver.

$$\underset{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}{<span\; style="font-family:\; inherit;">\iiint </span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">,</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">,</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">=</span>\underset{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}{<span\; style="font-family:\; inherit;">\iiint </span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">sin</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>\mathrm{<span\; style="font-family:\; inherit;">cos</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span><span\; style="font-family:\; inherit;">,</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">sin</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>\mathrm{<span\; style="font-family:\; inherit;">sin</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span><span\; style="font-family:\; inherit;">,</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">cos</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span><span\; style="font-family:\; inherit;">)</span>\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}^{\mathrm{<span\; style="font-family:\; inherit;">2</span>}}\mathrm{<span\; style="font-family:\; inherit;">sin</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}$$

Exemplo de Aplicação:

Vamos agora aplicar esses conceitos a um exemplo prático. Suponha que queremos calcular o volume de uma esfera de raio

$�$. Usando coordenadas esféricas, podemos definir nossa função $�(�,�,�)$ como $1$, já que estamos simplesmente interessados no volume. A região $�$ será a esfera de raio $�$

Então, nossa integral tripla se torna:

$$\underset{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}{<span\; style="font-family:\; inherit;">\iiint </span>}\mathrm{<span\; style="font-family:\; inherit;">1</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">=</span>\underset{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}{<span\; style="font-family:\; inherit;">\iiint </span>}{\mathrm{<span\; style="font-family:\; inherit;">�</span>}}^{\mathrm{<span\; style="font-family:\; inherit;">2</span>}}\mathrm{<span\; style="font-family:\; inherit;">sin</span>}<span\; style="font-family:\; inherit;"></span><span\; style="font-family:\; inherit;">(</span>\mathrm{<span\; style="font-family:\; inherit;">�</span>}<span\; style="font-family:\; inherit;">)</span>\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\text{<span style="font-family: inherit;">\hspace{0.17em}</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}\mathrm{<span\; style="font-family:\; inherit;">�</span>}$$

Integrando isso ao longo dos limites apropriados, encontramos o volume da esfera.

Conclusão:

E aqui está! Completamos nossa jornada através das integrais triplas em coordenadas esféricas. Espero que agora você tenha uma compreensão mais profunda deste tópico e possa aplicá-lo em seus próprios estudos e problemas.

Agora compartilhe o que você aprendeu, porque APRENDER É A NOSSA MELHOR HABILIDADE!

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