   Chapter 15, Problem 28RE

Chapter
Section
Textbook Problem

Calculate the value of the multiple integral.28. ∬ D x   d A , where D is the region in the first quadrant that lies between the circles x2 + y2 = 1 and x2 + y2 = 2

To determine

To calculate: The value of given double integral over the region R.

Explanation

Given:

The region D in the first quadrant lies between the circles x2+y2=1 and x2+y2=2 .

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:

Since, the equation of the circle is given, it is more appropriate to use polar coordinates. From the given conditions, it is observed r varies from 1 to 2 and θ varies from 0 to π2

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