Explanation The polar coordinates are: x = r cos θ , y = r sin θ , d x d y = r d r d θ The integration of f ( x , y ) over the region D can be calculated by: ∫ ∫ D f ( x , y ) d A = ∫ θ 1 θ 2 ∫ r = r 1 ( θ ) r 2 ( θ ) f ( r cos θ , r sin θ ) r d r d θ Given: f ( x , y ) = y ( x 2 + y 2 ) − 1 ; y ≥ 1 2 , x 2 + y 2 ≤ 1 Calculation: Since, the polar coordinates are: x = r cos θ , y = r sin θ , d x d y = r d r d θ And y = 1 2 Then r sin θ = 1 2 1 2 sin θ = r 1 2 sin θ = 1 since x 2 + y 2 = 1 ; r = 1 sin θ = 1 2 sin θ = sin π 6 or sin 5 π 6 θ = π 6 or 5 π 6 Now, the limits of the region D is: D = { ( r , θ ) | 1 2 sin θ ≤ r ≤ 1 , π 6 ≤ θ ≤ 5 π 6 } Now, the sketch of the region is as shown: Now, evaluating ∬ D y ( x 2 + y 2 ) − 1 d A , ∬ D y ( x 2 + y 2 ) − 1 d A = ∬ D y x 2 + y 2 d A = ∫ θ = π 6 5 π 6 ∫ r = 1 2 sin θ 1 ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.4, Problem 5E

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The integration of $f (x, y)$ over D using polar coordinates and sketch the region D.

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Chapter 16.4, Problem 4E

Chapter 16.4, Problem 6E

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Chapter 16 Solutions

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The integration of f ( x , y ) over D using polar coordinates and sketch the region D .

Solution Summary: The author calculates the integration of f (x, y) over the region D using polar coordinates.

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The integration of $f (x, y)$ over D using polar coordinates and sketch the region D.

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Chapter 16 Solutions

The integration of f ( x , y ) over D using polar coordinates and sketch the region D .

Solution Summary: The author calculates the integration of f (x, y) over the region D using polar coordinates.

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To calculate: The integration of f(x,y)over D using polar coordinates and sketch the region D.

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Chapter 16 Solutions

To calculate:

The integration of $f (x, y)$ over D using polar coordinates and sketch the region D.