   Chapter 15, Problem 60RE

Chapter
Section
Textbook Problem

(a) Evaluate ∬ D 1 ( x 2 + y 2 ) n / 2 d A , where n is an integer and D is the region bounded by the circles with center the origin and radii r and R, 0 < r < R.

(a)

To determine

To evaluate: The given integral.

Explanation

Given:

The function, f(x,y)=1(x2+y2)n2.

The region D lies between the circles centered at origin and the radii r and R, respectively.

Formula used:

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)

Calculation:

The radius of the smaller circle is denote the letter r. so, use the variable t for the polar coordinates instead of r. From the given region D, it is observed that the value of t varies from r to R and the value of θ varies from 0 to 2π. Substitute x=tcosθ and y=tsinθ

(b)

To determine

To find: For what values of the integer n, the integral in part (a) have a limit as r0+.

(c)

To determine

To evaluate: The given integral.

(d)

To determine

To find: For what values of the integer n, the integral in part (c) have a limit as r0+.

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