Suppose we want to determine the volume of a sphere with radius \(R\). Spherical coordinates are clearly the best option so the integral we are interested in is,

\[ \iiint_D \rho^2 \sin(\phi)\ d\rho\ d\phi\ d\theta \,.\]By Fubini's theorem, we can change around the order of variables as we like, as long as we are consistent with the limits of integration which are given as,

\[\begin{aligned} 0 \leq\ &\rho\ \leq R \\ 0 \leq\ &\theta\ \leq 2\pi \\ 0 \leq\ &\phi\ \leq \pi/2 \\ \end{aligned}\]and then we'll need to double it since we're only integrating over the top hemisphere.

```
using SymPy
@vars ρ θ ϕ R
intlims = Dict{Sym, Tuple{Int64, Number}}(
ρ => (0, R),
θ => (0, 2π),
ϕ => (0, π/2)
)
eqn = ρ^2 * sin(ϕ)
2 * integrate(eqn,
(ρ, intlims[ρ][1], intlims[ρ][2]),
(ϕ, intlims[ϕ][1], intlims[ϕ][2]),
(θ, intlims[θ][1], intlims[θ][2])
)
> 4.18879020478639𝑅3
```

Which is approximately equal to the known volume of a sphere,

\[ \frac{4}{3}\pi R^3 \,.\]
© Daniel Marvin. Last modified: July 19, 2021.