   Chapter 15.3, Problem 14E

Chapter
Section
Textbook Problem

Evaluate the given integral by changing to polar coordinates.14. ∬ D x   d A , where D is the region in the first quadrant that lies between the circles x2 + y2 = 4 and x2 + y2 = 2x

To determine

To evaluate: The given integral by changing into the polar coordinates.

Explanation

Given:

The function, f(x,y)=x .

The region D is lies between the circles x2+y2=4 and x2+y2=2x in the first quadrant.

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:

From the given region D, it is observed that the value of r varies from 0 to 2 and the value of θ varies from 0 to π2 . The region D lies between the two given curves.

Substitute x=rcosθ and y=rsinθ in the equation (1),

DxdA=x2+y24xdAx2+y22xxdA=x2+y24xdA(x1)2+y21xdA=0π202rcosθ(r)drdθ0π202cosθrcosθ(r)drdθ=0π202r2cosθdrdθ0π202cosθr2cosθdrdθ

Integrate the function with respect to θ and r by using the equation (2).

0π202r2cosθdrdθ0π202cosθr2cosθdrdθ=02r2dr0π2cosθdθ0π202cosθr2cosθdrdθ=[r33]02[sinθ]0π20π2[r3cosθ3]02cosθdθ=[233

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