   Chapter 15.3, Problem 13E

Chapter
Section
Textbook Problem

Evaluate the given integral by changing to polar coordinates.13. ∬ R arctan ( y / x )   d A , where R = {(x, y) | 1 ≤ x2 + y2 ≤ 4, 0 ≤ y ≤ x}

To determine

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

Explanation

Given:

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

The region is R={(x,y)|1x2+y24,0yx} .

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 R, it is observed that the value of r varies from 0 to 1 and the value of θ varies from 0 to π4 .

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

Rarctan(yx)dA=0π412arctan(rsinθrcosθ)(r

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