Explanation Given: f ( x , y ) = x 2 + y 2 ; 1 ≤ x 2 + y 2 ≤ 4 Formula: If f is continuous on the polar region D = { ( r , θ ) | α ≤ θ ≤ β , h 1 ( θ ) ≤ r ≤ h 2 ( θ ) , t h e n ∫ ∫ D f ( x , y ) d A = ∫ α β ∫ h 1 ( θ ) h 2 ( θ ) f ( r cos θ , r sin θ ) r d r d θ Since the radius is between 1 and 4 , therefore the co-ordinates lie in first quadrant. In polar co-ordinates, D is given by 1 ≤ r ≤ 2 , 0 ≤ θ ≤ π 2 Moreover, the integrand f ( x , y ) = x 2 + y 2 after changing the integrals becomes f ( r , θ ) = x 2 + y 2 = (rcos θ ) 2 + ( r sin θ ) 2 = r 2 cos 2 θ + r 2 sin 2 θ ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.4, Problem 2E

To determine

Sketch of Region D indicated and integrate $f (x, y)$ over D using polar co-ordinates.

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Chapter 16.4, Problem 1E

Chapter 16.4, Problem 3E

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Chapter 16 Solutions

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Sketch of Region D indicated and integrate f ( x , y ) over D using polar co-ordinates.

Solution Summary: The author explains how to determine the integral over D using polar co-ordinates.

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Sketch of Region D indicated and integrate $f (x, y)$ over D using polar co-ordinates.

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Chapter 16 Solutions

Sketch of Region D indicated and integrate f ( x , y ) over D using polar co-ordinates.

Solution Summary: The author explains how to determine the integral over D using polar co-ordinates.

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Sketch of Region D indicated and integrate f(x,y)over D using polar co-ordinates.

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Chapter 16 Solutions

Sketch of Region D indicated and integrate $f (x, y)$ over D using polar co-ordinates.