Question

calcule a integral tripla a seguir usando coordenadas cilíndricas:
///w x^3+xy^2 dV, com W sendo a região do primeiro octante abaixo de z=1-x^2-y^2.

Answer 1

Representación de la región G
          $G=\{(r,\theta,z): 0\ \textless \ r\ \textless \ 1\;,\;0\ \textless \ \theta \ \textless \ \pi/2\;,\;0\ \textless \ z\ \textless \ 1-r^2\}$

$\displaystyle V=\iiint_G r\cos \theta (r^2)r\,dr\,d\theta \,dz\\ \\ V=\int_{0}^1\int_{0}^{\pi/2}\int_{0}^{1-r^2}r^4\cos \theta dz\, d\theta \, dr\\ \\ V=\int_{0}^1r^4\int_{0}^{\pi/2}(1-r^2)\cos\theta\,d\theta\,dr\\ \\ V=\int_{0}^1r^4(1-r^2)\int_{0}^{\pi/2}\cos \theta d\theta\,dr\\ \\ V=\int_{0}^1r^4-r^6 dr\\ \\ V=\left.\left(\dfrac{r^5}{5}-\dfrac{r^7}{7}\right)\right|_{0}^1\\ \\ \boxed{V=\dfrac{2}{35}}$