# Find the average value of the function f ( x , y ) = 9 ? x 2 + y 2 on the annular region a 2 ?...

## Question:

Find the average value of the function {eq}f(x, y) = \frac{9}{\sqrt{x^2 + y^2}} {/eq} on the annular region {eq}a^2 \leq x^2 - y^2 \leq b^2 {/eq}, where {eq}0 < a < b {/eq}.

{eq}f_{ave} = \:? {/eq}

## Average Value of the Function:

The average value of any function is limit of function approaches to infinity.

Let f(x) be any function on the interval {eq}(a,b) {/eq}, which lies within the domain of the function, then the formula of the average value of the function is written as,

{eq}\dfrac{{\int {\int {f\left( {x,y} \right)dxdy} } }}{{\int {\int {dA} } }} {/eq}

## Answer and Explanation:

**Given**

- The function is given by,

{eq}f\left( {x,y} \right) = \dfrac{9}{{\sqrt {{x^2} + {y^2}} }} {/eq}.

- The annular region is {eq}{a^2} \le {x^2} +...

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