Consider the [[Integration in Cartesian coordinates|triple integral]] $\int \int \int_{\Omega} f \, dV$ where $\Omega \subseteq R^{3}$ is the volume under a [[Algebraic Surface|surface]]in $\mathbb{R}^{3}$. Suppose $\Omega$ has spherical symmetry, then instead of using *cartesian coordinates*, we might consider using [[Spherical Coordinates of Element of Real 3-Space|spherical coordinates]] $(r,\theta,\phi)$. Note: $dV=r^{2}\sin \phi \,dr \,d\theta \,d\phi$. # Examples > [!Example] Example (Volume of a sphere) > Find the volume of a sphere radius $a$. > >**Solution** $\begin{align} V &=\int_{\phi=0}^{\pi} \int_{\theta=0}^{2\pi} \int_{0}^{a} r^{2} \sin \phi \, dr \, d\theta \, d\phi \\ & = 2\pi \int_{0}^{\pi} \sin \phi \, d\phi \; \int_{0}^{a}r^ 2 \, dr \\ & = 2\pi \cdot 2 \cdot \frac{a^{3}}{3} = \frac{4\pi a^{3}}{3} \end{align}$