• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Integral over a sphere

  • Thread starter rocket
  • Start date
10
0
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].

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"
 

HallsofIvy

Science Advisor
Homework Helper
41,683
865
Okay, what have you done on this?
 

Related Threads for: Integral over a sphere

Replies
10
Views
2K
  • Posted
Replies
17
Views
19K
W
Replies
2
Views
2K
  • Posted
Replies
1
Views
2K
Replies
23
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top