# Calculate the mean of a function within a part of a circle

I have a cellular signal calculation function, which calculates the signal given the distance from the antenna. Without the constants, the function is basically: $$f(d)=1/(d^α)$$ where $$α$$ is a parameter.

I need to find the mean of a part of a circle containing points which are the function values. I will show on an example:

In this picture, assume the center of function $$f$$ is the center of the yellow circle. Each point inside the green circle has distance $$d$$ from the center, and a value $$f(d)$$.

I want to find the mean of the values for all the points in the green circle(excluding the blue area, which is determained by the yellow circle!)

I also want to find the mean of the values for all the points in the blue circle area, which I assume is done by substracting.

I saw here that it can be done for a full circle. But how to do it for a part of a circle?

• Does the green circle pass through the center of the yellow circle?
– Jens
Commented Oct 7, 2018 at 10:10
• @Jens Not necessarily. Basically I need to compute the average function value over a part of a circle. Commented Oct 7, 2018 at 19:19
• @Amitai Yuval can you please try to help? Commented Oct 10, 2018 at 14:26

Let $$O$$ be the center of the yellow disc and $$\rho_0$$ its radius. Let the blue disc have its center at distance $$R$$ from $$O$$, and let $$\rho$$ be its radius. Consider now concentric circles $$\gamma_r$$ of variable radius $$r$$ centered at $$O$$. For given $$r\in[R-\rho,R+\rho]$$ find by trigonometric reasoning the angle $$2\alpha(r)$$ of the arc that the blue disc cuts out from $$\gamma_r$$. Then the integrals $$\int_{R-\rho}^{\rho_0} f(r)\>2\alpha(r)\>r\>dr,\qquad \int_{\rho_0}^{R+\rho} f(r)\>2\alpha(r)\>r\>dr$$ give the "total signal energy" received in the two parts of the blue disc. Divide by the corresponding areas, and you obtain the average signal strengths.