Explanation Given: f ( x , y ) = e − ( x 2 + y 2 ) And region of integration is disk defined by x 2 + y 2 ≤ a 2 Concept Used: ∬ D f ( x , y ) d A = ∬ D f ( r cos θ , r sin θ ) r d r d θ The conversion from cartesian coordinates to polar coordinates is given by: x = r cos θ , y = r sin θ , x 2 + y 2 = r 2 , d y d x = r d r d θ Calculation: To find ∬ D f ( x , y ) d A Where D is the disk defined by x 2 + y 2 ≤ a 2 and f ( x , y ) = e − ( x 2 + y 2 ) Change the above integral in polar coordinates, as ∬ D f ( x , y ) d A = ∬ D f ( r cos θ , r sin θ ) r d r d θ = ∬ D e − r 2 r d r d θ , since f ( x , y ) = e − ( x 2 + y 2 ) Now, let’s define the region of the integration, as the region is a whole disk defined by x 2 + y 2 ≤ a 2 This denotes a circle of radius a centered at origin. Thus r varies from 0 to a , and the angle θ will vary from 0 to 2 π . So, the limit of integration in polar coordinates would be 0 ≤ r ≤ a , and 0 ≤ θ ≤ 2 π So, the given double integral in polar coordinates can be calculated as ∬ D e − r 2 r d r d θ = ∫ 0 2 π ∫ 0 a e − r 2 r d r d θ To solve the inner integral use substitution as, Put r 2 = t ⇒ 2 r d r = d t ⇒ r d r = d t 2 Now change limits as

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 12.3, Problem 28PS

To determine

To find

∬Df(x,y)dA

Where D is the disk defined by x2+y2≤a2

And f(x,y)=e−(x2+y2)

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Chapter 12.3, Problem 27PS

Chapter 12.3, Problem 29PS

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Chapter 12 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

Ch. 12.1 - Prob. 1PS Ch. 12.1 - Prob. 2PS Ch. 12.1 - Prob. 3PS Ch. 12.1 - Prob. 4PS Ch. 12.1 - Prob. 5PS Ch. 12.1 - Prob. 6PS Ch. 12.1 - Prob. 7PS Ch. 12.1 - Prob. 8PS Ch. 12.1 - Prob. 9PS Ch. 12.1 - Prob. 10PS

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To find ∬ D f ( x , y ) d A Where D is the disk defined by x 2 + y 2 ≤ a 2 And f ( x , y ) = e − ( x 2 + y 2 )

Solution Summary: The author explains how to change the above integral in polar coordinates.

Videos

To find

∬Df(x,y)dA

Where D is the disk defined by x2+y2≤a2

And f(x,y)=e−(x2+y2)

Want to see the full answer?

Chapter 12 Solutions

To find ∬ D f ( x , y ) d A Where D is the disk defined by x 2 + y 2 ≤ a 2 And f ( x , y ) = e − ( x 2 + y 2 )

Solution Summary: The author explains how to change the above integral in polar coordinates.

Videos

To find ∬Df(x,y)dA Where D is the disk defined by x2+y2≤a2 And f(x,y)=e−(x2+y2)

Want to see the full answer?

Chapter 12 Solutions

To find

∬Df(x,y)dA

Where D is the disk defined by x2+y2≤a2

And f(x,y)=e−(x2+y2)