Register to reply 
Average of COSIN 
Share this thread: 
#1
Aug3113, 05:24 PM

P: 8

hi,
how can I calculate average of cos^{2}x ? I want to take average over a sphere I tried to do like this: <cos^{2}X>= 1/2π ∫cos^{2}xdx and I get 1/2 but in my books, wrote that average of cos^{2}x , taken over a sphere, is 1/3 


#2
Aug3113, 05:38 PM

Emeritus
Sci Advisor
PF Gold
P: 4,500

What sphere are you trying to average it over?
1/2 is the average of cos^{2}(x) on the interval [0,2pi], which is something that nobody would call a sphere. 


#3
Sep113, 02:30 AM

P: 8

in fact, my question is  how can I take average over sphere?..



#4
Sep113, 03:34 AM

P: 128

Average of COSIN
What's x, is it something "specific"? Because if it happens to be, for example, the polar angle in spherical coordinates, then my guess is that you're supposed to calculate a surface integral over a sphere, [itex] A^{1} \iint_A \cos^2(\theta) \mathrm{d}A[/itex], where A is the surface area of a sphere and dA is the area element. The radius will cancel out. This gives you the correct answer, but it could obviously be something else as well. But, like Office_Shredder said, just integrating over the interval [0,2π] won't do, you're certainly not taking the average over a sphere that way.



#5
Sep113, 07:04 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,503

To further confuse things, your function, cos(x) depends only on a single variable, x. Is that what you intended or did you mean to have a function of all three variables, x, y, and z or in polar coordinates, [itex]\theta[/itex] and [itex]\phi[/itex]? And do you mean the three dimensional ball or the surface of the sphere. The volume of a ball of radius R is [itex](4/3)\pi R^3[/itex] while the surface area is [itex]4\pi R^2[/itex]. 


#6
Sep113, 07:52 AM

P: 8

we have a function:
G(t)=cos(x)^2+sin(x)^2*cos(wt) X is angle between two vector, if the vectors direction is random, then averaging over all directions would be yield G(t) = 1/3 +2/3*cos(wt) /////////// I just do not understand, how to get it :) 


#7
Sep113, 08:34 AM

P: 97

<cos^{2}(x)>= 1/2 ∫cos^{2}(x) sin(x) dx and with appropriate limits... === and similar with sinus if necessary ==== edit; I thought it was in the HW section 


Register to reply 
Related Discussions  
Ambiguity in forumlas for average speed & average velocity  Classical Physics  6  
Why is Cosin and Secant even?  Precalculus Mathematics Homework  7  
Simple average speed/average velocity problem  Introductory Physics Homework  2  
Word problem: Asking to solve for average velocity and average speed  Introductory Physics Homework  5  
Sin and Cosin law  Precalculus Mathematics Homework  7 