Inetgral mean for a function of 2 variables

[Methavix]

Homework Statement

I have a function of two variables. I want to yield the integral mean over these 2 variables.
In particular, the function is a vector (of constant length v0) coming out from a only point that can have all direction possible in a semi-space. So if i draw all vectors i draw a semi-sphere of radius v0. I call alpha one angle and beta the other one, and both of them are variable between 0 and pi.

2. The attempt at a solution

If a consider just a 2-dimension space i have the solution

....pi
..../
v = (1/pi) * | v0*sin(alpha)*d(alpha) = (2/pi)*v0
.../
...0

but in the 3-D space? which is the mean vector? (that is horizontal, i mean with alpha=beta=0).

thanks
Luca

 

[mighty2000]

Hello Luca,

Thank you for your post. To find the integral mean over two variables, you can use the formula:

mean = (1/A) * ∫∫ f(x,y) dxdy

where A is the area of the region over which the mean is being taken and f(x,y) is the function of two variables.

In your case, the region is a semi-sphere of radius v0, which has an area of 2πv0^2. The function is a vector of constant length v0, so we can rewrite the integral as:

mean = (1/2πv0^2) * ∫∫ v0 sin(α) dαdβ

To solve this integral, we can use the fact that the integral of sin(α) over the range [0,π] is equal to 2. Therefore, the integral becomes:

mean = (1/2πv0^2) * 2 * π * 2 = 4/πv0^2

This means that the mean vector in the 3-D space is a vector of length 4/πv0^2 in the horizontal direction. I hope this helps. Let me know if you have any further questions.