   Chapter 15.3, Problem 37E

Chapter
Section
Textbook Problem

Find the average value of the function f ( x , y )   =   1 / x 2 + y 2 on the annular region a2 ≤ x2 + y2 ≤ b2 where 0 < a < b.

To determine

The average value of the function on the given annular region.

Explanation

Formula used:

The average value of given function over the region D is,

fave=1A(D)Df(x,y)dA, where A(D) is the area of the given region D.

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π, then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Given:

The function is, f(x,y)=1x2+y2.

The region D is the annular region a2x2+y2b2.

Calculation:

Substitute x=rcosθ,y=rsinθ in the given conditions. So, the region D becomes,

a2r2b2arb

And θ varies from 0 to 2π. Since it is observed from the given conditions that the region D is a annular region, the area of the given region is,

Area=πr12πr22=πb2πa2

Substitute x=rcosθ,y=rsinθ and convert the given function into polar coordinates

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