Question

Alguém poderia me ajudar nessa questão de calculo?

Answer 1

Transformación
$x=r\cos \theta \sin \phi\\ y=r\sin \theta \sin \phi\\ z=r\cos \phi$

donde
              $\theta \in [0,2\pi )\;,\; \phi \in [0,\pi/2]\;,\; r\in (0,1]$

El valor absoluto del Jacobiano es:
           $\left|\dfrac{\partial(x,y,z)}{\partial(r,\theta,\phi)}\right|=r^2|\sin\phi|$

Cambio de variable

          $\displaystyle \iiint\limits_{V} \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz=\iiint\limits_{H}r\cdot r^2|\sin \phi|\,dr\,d\theta\,d\phi\\ \\ \\ I=\int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{0}^1 r^3|\sin \phi| dr \,d\phi\, d\theta\\ \\ I=2\pi\int_{0}^{\pi/2}|\sin \phi|\,d\phi\int_{0}^1 r^3 dr \\ \\ I=2\pi \int_{0}^{\pi/2}\sin \phi\,d\phi \cdot \dfrac{1}{4}\\ \\ \\ \boxed{I=\dfrac{\pi}{2}}$