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 [latex]50[/latex] ft, using the equation [latex]x^2+y^2+z^2=r^2[/latex].

Solution

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

[latex]0\le{\varphi}\le{\frac{\pi}{2}},0\le{\rho}\le{r,0}\le\theta\le\frac{\pi}{2}[/latex]Therefore,

[latex]\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}}[/latex]

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