A sphere, centered at the origin, has radius 2. Find integrals that compute its volume, using Cartesian, cylindrical, and spherical coordinates. For your answers $\theta =$ theta, $\phi =$ phi, and $\rho =$ rho.

Cartesian

$\displaystyle\int_a^b\int_c^d\int_e^f p(x,y,z) \,dz\,dx\,dy =$

$\displaystyle\int$

$\displaystyle\int$

$\displaystyle\int$

$\hspace{5pt}dz\,dx\,dy$

Cylindrical

$\displaystyle\int_a^b\int_c^d\int_e^f f(r,\theta,z) \,dz\,dr\,d\theta =$

$\displaystyle\int$

$\displaystyle\int$

$\displaystyle\int$

$\hspace{5pt}\,dz\,dr\,d\theta$

Spherical

$\displaystyle\int_a^b\int_c^d\int_e^f g(\rho,\theta,\phi) \,d\rho \,d\theta \,d\phi =$

$\displaystyle\int$

$\displaystyle\int$

$\displaystyle\int$

$\hspace{5pt}d\rho \,d\theta\ \,d\phi$