Question

Como resolver essa integral dupla

Answer 1

Olá


$\displaystyle \mathsf{ \int\limits^2_0 \int\limits^{ \sqrt{4-y^2}}_ 0 { \frac{2}{ \sqrt{4-y^2} } } \, dxdy }\\\\\\\\\text{Integrando em dx, y se torna constante}\\\\\\ \mathsf{ \int\limits^2_0 \left[\int\limits^{ \sqrt{4-y^2}}_ 0 { \frac{2}{ \sqrt{4-y^2} } } \, dx\right]dy }\\\\\\ \mathsf{ \int\limits^2_0 \left[ \left({ \frac{2}{ \sqrt{4-y^2} }\cdot x } \right)\bigg|^{^{ \sqrt{4-y^2}}}_0\, \right]dy }$

$\displaystyle \mathsf{\int\limits^2_0 \left[ \left(\frac{2}{\sqrt{4-y^2}}\cdot (\sqrt{4-y^2})\right)~-~\left(\frac{2}{\sqrt{4-y^2}}\cdot (0})\right) \right]dy }$

$\displaystyle \mathsf{ \int\limits^2_0 {2} \, dy }\\\\\\\mathsf{=2y\bigg|^2_0}\\\\\\\mathsf{=2\cdot (2)~-~2\cdot (0)}\\\\\\\boxed{\mathsf{=4}}$


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