let [tex]B_n(r) = \{x \epsilon R^n| |x| \le r\} [/tex] be the sphere around the origin of radius r in [tex] R^n. [/tex] let [tex] V_n(r) = \int_{B_n(r)} dV [/tex] be the volume of [tex]B_n(r)[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

a)show that [tex] V_n(r) = r^n * V_n(1) [/tex]

b)write [tex] B_n(1) [/tex] as [tex] I*J(x) * B_{n-2}(x,y), [/tex] where I is a fixed interval for the variable x, J an interval for y dependent on x, and [tex]B_{n-2}(x,y) [/tex] a ball in [tex] R^{n-2} [/tex] with a radius dependent on x and y. this decomposition should allow for use of fubini's theorem in part c)

c)find [tex]V_n(1) [/tex] in terms of [tex] V_{n-2}(1)[/tex]

d)find [tex] V_n(1) [/tex] in terms of only n (eg. find a closed form for [tex] V_n(1) [/tex])

for b), is the answer

[tex] 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]\} [/tex]

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 [tex] V_n(1) = V_{n-2}(1) * \int_{0}^{2 \pi} \int_{0}^{1} r dr d{\theta} [/tex] (using polar coordinates)

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Integral over a sphere

**Physics Forums | Science Articles, Homework Help, Discussion**