Question

Segue integral montada e respondida.

Answer 1

$\int_{0}^{\pi}\int_{0}^{2\pi}\int_{0}^{\sqrt{5}} \rho^2*sen(\phi)\;d\rho\; d\theta \;d\phi\\\\ = \int_{0}^{\pi}sen(\phi) d\phi *\int_{0}^{2\pi}d\theta* \int_{0}^{\sqrt{5}} \rho^2 d\rho\\\\= \left[-cos(\phi) \right]_0^{\pi} * \left[ \theta \right]_0^{2\pi} * \left[ \frac{\rho^3}{3} \right]_0^{\sqrt{5}}\\\\ = [-cos(\pi)-cos(0)] * [2\pi-0]* [ \frac{(\sqrt{5})^3}{3}- \frac{0^3}{3} ]\\\\ =[-(-1)-1]*[2\pi]*[ \frac{(\sqrt{5})^3}{3} ]\\\\ = 2*2\pi * \frac{5\sqrt{5}}{3} \\\\ = \frac{20\pi\sqrt{5}}{3}$