TITLE: Volume of a sphere by "adding" half-spheres of lower dimension
QUESTION [10 upvotes]: I'm wondering about different ways to compute the volume of an $n$-sphere.  Please see the wikipedia page for one method to compute the volume via hyperspherical coordinates:
http://en.wikipedia.org/wiki/N-sphere#Hyperspherical_volume_element
Suppose now I want to compute the volume $V(n)$ of an $n$-sphere by integrating the volumes $V(n-1)$ of a whole bunch of $(n-1)$-spheres.  Assuming that this $(n-1)$-sphere is aligned with the first $n-1$ coordinate axes, I don't see a way to just integrate $(n-1)$-spheres with the variable being the last angular coordinate.  I'm also more generally interested in this question when we integrate $(n-k)$-spheres with volumes $V(n-k)$ with the variables being the last $k$  angular coordinates.  Any thoughts?

REPLY [2 votes]: First, consider integrating for the volume of a circle along a line.
$$2\int_0^r (r^2-x^2)^.5dx$$
This one's somewhat involved, as is every subsequent integral that yields an increase in pi power, because you have to use polar on the square root, but you eventually get:
$$A=\pi r^2$$
Now, we want to find the volume of a sphere by adding up the volumes of circular cross sections of the sphere along the x-axis. We already know the area of a circle because we calculated it from the previous integral. The radius of the circle is sqrt(r^2-x^2) and there is symmetry, so the integral is
$$ 2\pi\int_0^r r^2-x^2dx$$
$$ 2\pi r^3-2\pi r^3/3$$
$$ 4\pi r^3/3$$
You just continue.
$$4\pi/3\int_0^r (r^2-x^2)^{1.5}dx$$
and so forth for higher dimension hyperspheres.
The OP said something about an angle. If you want to integrate with respect to an angle, change to polar/spherical and follow the same method.