Finding the Volume of l’Hemisphèric

Find the volume of the spherical planetarium in l’Hemisphèric in Valencia, Spain, which is five stories tall and has a radius of approximately $50$ ft, using the equation $x^2+y^2+z^2=r^2$.

Solution

We calculate the volume of the ball in the first octant, where $x\ge{0}, y\ge{0}, \text{ and } z\ge{0}$, using spherical coordinates, and then multiply the result by $8$ for symmetry. Since we consider the region $D$ as the first octant in the integral, the ranges of the variables are

$0\le{\varphi}\le{\frac{\pi}{2}},0\le{\rho}\le{r,0}\le\theta\le\frac{\pi}{2}$Therefore,

$\hspace{5cm}\large{\begin{align} V&=\underset{D}{\displaystyle\iiint}dx \ dy \ dz = 8\displaystyle\int_{\theta=0}^{\theta=\pi/2}\displaystyle\int_{\rho=0}^{\rho=\pi} \rho^{2}\sin\theta d\varphi d\rho d\theta \\ &=8\displaystyle\int_{\varphi=0}^{\varphi=\pi/2} \ d\varphi\displaystyle\int_{\rho=0}^{\rho=r}\rho^2 \ d\rho\displaystyle\int_{\theta=0}^{\theta=\pi/2}\sin\theta \ d\theta \\ &=8\left(\frac{\pi}2\right)\left(\frac{r^3}3\right)(1) \\ &=\frac43\pi{r}^3 \end{align}}$

This exactly matches with what we knew. So for a sphere with a radius of approximately $50$ ft, the volume is $\frac{4}{3}\pi(50)^3\approx 523,600$ ft³.