Question

integral tripla de xy^2 , onde G é o conjunto de pontos que satisfaz -1 x 1, 0 y 1 e 0 z pi/2.

Answer 1

           $\displaystyle I=\int _{-1}^{1}\int _{0}^{1}\int _{0}^{\pi/2}xy^2\, dz\,dy\,dx\\ \\ \\ I=\int _{-1}^{1}\int _{0}^{1}\left.(xy^2z)\right|_{z=0}^{z=\pi/2}\,dy\,dx\\ \\ \\ I=\int _{-1}^{1}\int _{0}^{1}\dfrac{\pi}{2}\cdot xy^2\,dy\,dx$


           $\displaystyle I=\dfrac{\pi}{2}\int _{-1}^{1}\int _{0}^{1}xy^2\,dy\,dx\\ \\ \\ I=\dfrac{\pi}{2}\int _{-1}^{1}\left.\left(x\cdot \dfrac{y^3}{3}\right)\right|_{y=0}^{y=1}\,dx\\ \\ \\ I=\dfrac{\pi}{2}\int _{-1}^{1}\left(x\cdot \dfrac{1}{3}\right)\,dx$
 
           $\displaystyle I=\dfrac{\pi}{6}\int _{-1}^{1}x\,dx\\ \\ \\ I=\dfrac{\pi}{6}\left.\left(\dfrac{x^2}{2}\right)\right|_{-1}^1\\ \\\\ \boxed{I=0}$