integral dupla
Enviado: Seg 15 Abr, 2019 15:19
Resolvendo a integral dupla, [tex3]\int\limits_{0}^{}\frac{\pi }{2}\int\limits_{0}^{\sen 0} ( r.\cos0) dr d0, obtém-se?[/tex3]
Olá Hully,
Posso estar errado, mas não seria:
[tex3]\int\limits_{0}^{\frac{\pi}{2}}\int\limits_{0 }^{\sen \theta} ( r \cdot \cos\theta) \cdot dr \cdot d\theta[/tex3]
Caso seja assim, podemos fazer:
[tex3]\int\limits_{0}^{\frac{\pi}{2}} \cos \theta\int\limits_{0 }^{\sen \theta} r \cdot dr \cdot d\theta[/tex3]
Avaliando o integral:
[tex3]\int r = \frac{r^2}{2}[/tex3]
Devolvendo os limites:
[tex3]\int\limits_{0}^{\frac{\pi}{2}} \cos \theta \cdot \frac{r^2}{2} \Biggr |_{0}^{\sen \theta} d\theta[/tex3]
[tex3]\int\limits_{0}^{\frac{\pi}{2}} \frac{\cos \theta \cdot\sen^2 \theta}{2} \cdot d\theta[/tex3]
[tex3]\frac{1}{2} \cdot \int\limits_{0}^{\frac{\pi}{2}} \cos \theta \cdot\sen^2 \theta \cdot d\theta[/tex3]
Avaliando o integral:
[tex3]\int cos \theta \cdot\sen^2 \theta \cdot d\theta = \frac{\sen^3\theta}{3}[/tex3]
Devolvendo os limites:
[tex3]\frac{1}{2} \cdot \frac{\sen^3\theta}{3} \Biggr |_{0}^{\frac{\pi}{2}} [/tex3]
[tex3]\frac{1}{2} \cdot \frac{\sen^2 \left(\frac{\pi}{2}\right)}{3} - \frac{\sen^2 (0)}{3} [/tex3]
[tex3]\frac{1}{2} \cdot \frac{1}{3} [/tex3]
[tex3]\boxed{\frac{1}{6}}[/tex3]
