Explanation Explanation Given Figure showing the region D is given. The disc centered at (1,0) and radius 1 is excluded from a larger disc centered at origin and radius 2. Formula 1. Equation to transform Cartesian coordinates to polar coordinates x = r cos ( θ ) , y = r sin ( θ ) r d r d θ 2. Area element in polar coordinates . Calculation Transform the Cartesian coordinates to the polar coordinates. Then entire region inside the outer circle is given by the equations x = r cos ( θ ) , y = r sin ( θ ) where 0 ≤ r ≤ 2 , 0 ≤ θ ≤ 2 π . Then the transformed integral is given by ∬ E x 2 + y 2 d x d y = ∫ θ = 0 2 π ∫ r = 0 2 r × r d r d θ where E represent the entire region inside the outer circle. r d r d θ is the area element in polar coordinates. Calculate the integral to get ∬ E x 2 + y 2 d x d y = ∫ θ = 0 2 π ∫ r = 0 2 r × r d r d θ = ∫ θ = 0 2 π ( ∫ r = 0 2 r 2 d r ) d θ = 16 π 3 . Next we need to compute integral over the region inside the inner circle. The Cartesian representation of the inner circle is given by x = 1 + r cos ( ϕ ) , y = r sin ( ϕ ) where 0 ≤ r ≤ 1 , 0 ≤ ϕ ≤ 2 π . Any point in the circumference of the inner circle will be of the form x = 1 + cos ( ϕ ) , y = sin ( ϕ ) . The distance from origin to this point is ( 1 + cos ( ϕ ) ) 2 + sin ( ϕ ) 2 = 2 + 2 cos ( ϕ ) . Note that θ is the angle between the radius vector and the x-axis...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.4, Problem 23E

To determine

To find

The area integral $\iint_{D} \sqrt{x^{2} + y^{2}} d x d y$ where D is specified.

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Chapter 16.4, Problem 22E

Chapter 16.4, Problem 24E

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Find the area of the region in the polar coordinate plane . Inside the “figure eight” r = 1 + cos 2u and outside the circle r = 1

Chapter 16 Solutions

CALCULUS (CLOTH)

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To find The area integral ∬ D x 2 + y 2 d x d y where D is specified.

Solution Summary: The author explains how to find the area integral D x 2 + y 2 where D is specified.

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To find

The area integral $\iint_{D} \sqrt{x^{2} + y^{2}} d x d y$ where D is specified.

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Chapter 16 Solutions

To find The area integral ∬ D x 2 + y 2 d x d y where D is specified.

Solution Summary: The author explains how to find the area integral D x 2 + y 2 where D is specified.

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Chapter 16 Solutions

To find

The area integral $\iint_{D} \sqrt{x^{2} + y^{2}} d x d y$ where D is specified.