   Chapter 15.3, Problem 38E

Chapter
Section
Textbook Problem

Let D be the disk with center the origin and radius a. What is the average distance from points in D to the origin?

To determine

To find: The average distance from the points in the region D.

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 region D is the disk with center at origin and radius a

Calculation:

The distance between any point (x,y) to the origin is x2+y2.

Therefore, f(x,y)=x2+y2.

From the given region D, θ varies from 0 to 2π and r varies from 0 to a.

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

f(x,y)=x2+y2=r2=r

Since the region D is the disk with radius a, the area of the given disk is πa2

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