Integral over a sphere

1. Jul 19, 2005

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. this decomposition should allow for use of fubini's theorem in part c)

c)find $$V_n(1)$$ in terms of $$V_{n-2}(1)$$

d)find $$V_n(1)$$ in terms of only n (eg. find a closed form for $$V_n(1)$$)

$$B_n(1) = \{I, J \epsilon R^n, B_{n-2}(x,y) \epsilon R^{n-2} | I \epsilon [0,1], J \epsilon [-\sqrt{1-x^2}}, \sqrt{1-x^2}], B_{n-2}(x,y) \epsilon [0,1]*[0,1]*[0,1]\}$$
not sure about the last part...how do i show that radius of B_{n-2}(x,y) is dependent on x and y?

for c), i found the answer to be $$V_n(1) = V_{n-2}(1) * \int_{0}^{2 \pi} \int_{0}^{1} r dr d{\theta}$$ (using polar coordinates)

i'm stuck on d). what does it mean by "closed form"

2. Jul 19, 2005

HallsofIvy

Staff Emeritus
Okay, what have you done on this?