How Do You Derive the Volume of an n-Dimensional Sphere?

[rocket]

let B_n(r) = \{x \epsilon R^n| |x| \le r\} be the sphere around the origin of radius r in R^n. let V_n(r) = \int_{B_n(r)} dV be the volume of B_n(r).

a)show that V_n(r) = r^n * V_n(1)
b)write B_n(1) as I*J(x) * B_{n-2}(x,y), where I is a fixed interval for the variable x, J an interval for y dependent on x, and B_{n-2}(x,y) a ball in R^{n-2} with a radius dependent on x and y. set up an integral to allow for use of fubini's theorem in order to find V_n(1)in terms of V_{n-2}(1).


for a), I assume that V_n(r) is proportional to r^n. So V_n(r) = C*r^nwhere C is a constant. V_n(1) = C*(1)^n = C. we have the equation

V_n(1) / V_n(r) = C / C * r^n
V_n(1) / V_n(r) = 1 / r^n
V_n(r) = r^n * V_n(1)which completes the proof.

the only problem is, i don't know how to prove the assumption i used - that V_n(r) is proportional to r^n. I know that V_1(r) = 2 * r^1 = 2r, V_2(r) = \pi * r^2, and V_3(r) = 4/3 * \pi * r^3, which is how i guessed the assumption in the first place, but I don't know how to prove it holds true for V_n(r). I tried using induction but I don't know what is V_{n+1}(r) in terms of V_n(r). My instructor suggested that we set up an integral and use a change of variables of some sort. I was wondering how would I set up an integral to find the volume of a sphere in n-dimensions.


i'm having a lot of trouble understanding b). the bounds of the triple integral would be as follows: the interval for x would be [-1,1] for a sphere centered on the origin, since we're dealing with a radius of 1. the interval for y would be [\sqrt{1-x^2}, -\sqrt{1-x^2}]. But I don't understand how to derive the bounds for B_{n-2}(x,y). Also, how do we find what function over which to integrate?

 

[CarlB]

the only problem is, i don't know how to prove the assumption i used - that V_n(r) is proportional to r^n.

I assume you're defining V_n(r) as an integral over a volume, where r gives the radius. Fix n. Given r, you can recalculate the integral to be a constant times V_1(r) by making a variable change x_k <= r x_k' for 1 \leq k \leq n. The constant is your r^n factor.

Carl

 

[thepatientmental]

To prove that V_n(r) is proportional to r^n, we can use the formula for the volume of a sphere in n-dimensions, which is V_n(r) = \frac{\pi^{n/2}r^n}{\Gamma(n/2 + 1)}. This formula can be derived using integration and the change of variables method.

For b), we can use the formula for the volume of a sphere in n-dimensions and apply Fubini's theorem to set up an integral. We can write B_n(1) as the product of B_{n-2}(x,y) and the interval I*J(x), where I is the interval for x and J(x) is an interval for y dependent on x. The bounds for x would be [-1,1] and for y, we can use the equation \sqrt{1-x^2}, -\sqrt{1-x^2}. This can be derived using the Pythagorean theorem.

To find the function to integrate, we can use the formula for V_n(r) and substitute 1 for r. Then, we can use the change of variables method to convert the integral into a single variable integral in terms of x and y. This will give us an equation in terms of V_{n-2}(1), which we can solve for to find V_n(1) in terms of V_{n-2}(1).