Finding volume of a sphere using integration

Question

I have searched and found 2 methods of finding volume using integration :

• considering a small cylindrical element and integrating that over the radius
• considering a small circle element and using the relation x^2 + y^2 = r^2 and integrating it over the z-axis.

I was trying to find the integration by considering a small circle element (with radius r) and using the relation r = R cosθ where R is the radius of the sphere / hemisphere.

So I was thinking of calculating the volume of the hemisphere by integrating the π R^2 cos^2θ dθ from 0 to π/2. Is this method right? And how will the integration be like?

Answer 1

I do not like the "consider a small element" approach physicists often use, as it is not very intuitive and can easily produce errors. In this case you can just use the usual transformation of integrals: $\renewcommand{\phi}{\varphi}$

$$ \int_{\Omega} f(y) dy = \int_{\phi^{-1}(\Omega)} f(\phi(x)) |\det D\phi(x)| dx $$

Where $\varphi : \Omega \to \phi(\Omega) \subseteq \mathbb R^{n}$ is transformation of the coordinate systems and $D\phi : \Omega \to \mathbb R^{n\times n}$ is the corresponding Jacobi matrix.

So in your case all you have to do is set $f:\equiv 1$ and write down your transformation $\phi$. Then $|\det D\phi(x)|$ is the expression you are looking for, and the above equation simplifies to

$$Vol(\Omega) = \int_{\Omega} 1dy = \int_{\phi^{-1}(\Omega)} |\det D\phi(x)| dx $$

Answer 2

The best way of solving this problem it's using spherical coordinates. Then, you can only calculate the volume of one octant of the space supposing that the sphere is centered on the origin.

So, given a solid sphere with radius $R$, the volume would be:

$$ V = 8 \int_{0}^{\frac{\pi}{2}}{d\theta}\int_{0}^{\frac{\pi}{2}}{\int_{0}^{R}{r^{2}\sin{\varphi}\,\,dr}d\varphi} $$

Answer 3

Spherical co-ordinates are perfect for this

$$\int^{2\pi}_{0}\int^{\pi}_{0}\int^{r}_{0}r^2\cdot \sin(\phi)\cdot dr\cdot d\phi \cdot d\theta$$

you may find the following document an interesting read: https://sites.math.washington.edu/~aloveles/Math324Fall2013/f13m324TripleIntegralExamples.pdf