Explanation Calculation: Over the disk x 2 + y 2 ≤ 3 4 : Write the disk equation. x 2 + y 2 ≤ 3 4 Substitute r cos θ for x and r sin θ for y . ( r cos θ ) 2 + ( r sin θ ) 2 ≤ 3 4 r 2 cos 2 θ + r 2 sin 2 θ ≤ 3 4 r 2 ( cos 2 θ + sin 2 θ ) ≤ 3 4 r = 3 2 Considering the whole circle, the angle θ goes from 0 to 2 π . Determine the value of the integral function f ( x , y ) = 1 ( 1 − x 2 − y 2 ) . I = ∬ R 1 ( 1 − x 2 − y 2 ) d A Substitute r cos θ for x , r sin θ for y , and r d r d θ for d A . I = ∫ 0 2 π ∫ 0 3 2 1 ( 1 − r 2 cos 2 θ − r 2 sin 2 θ ) r d r d θ = ∫ 0 2 π ∫ 0 3 2 1 ( 1 − r 2 ) r d r d θ (1) Consider u = 1 − r 2 . Differentiate the term u with respect to r . d u d r = − 2 r − d u 2 = r d r Determine the limits. u = 1 − r 2 Substitute 0 for r . u = 1 Substitute 3 2 for r . u = 1 − ( 3 2 ) 2 = 1 − 3 4 = 1 4 Substitute u for 1 − r 2 and − d u 2 for r d r in Equation (1). I = ∫ 0 2 π ∫ 1 1 4 1 u ( − d u 2 ) d θ = 1 2 ∫ 0 2 π ∫ 1 1 4 ( − 1 u ) d u d θ = 1 2 ∫ 0 2 π [ − ln u ] 1 1 4 d θ = 1 2 ∫ 0 2 π [ − ln 1 4 + ln 1 ] d θ I = ln 4 2 [ θ ] 0 2 π = ln 4 2 ( 2 π ) = π ln 4 Therefore, the integral value of the function f ( x , y ) = 1 ( 1 − x 2 − y 2 ) is π ln 4 _

University Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780321999580

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14.4, Problem 43E

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Find the integral value of the function f(x,y)=1(1−x2−y2) over the disk x2+y2≤34.

Find whether the integral of function f(x,y) is exist over the disk x2+y2≤1 and provide reason for the result.

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Chapter 14.4, Problem 42E

Chapter 14.4, Problem 44E

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Chapter 14 Solutions

University Calculus: Early Transcendentals (3rd Edition)

Ch. 14.1 - In Exercises 1-14. evaluate the iterated...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - Prob. 10E

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Find the integral value of the function f ( x , y ) = 1 ( 1 − x 2 − y 2 ) over the disk x 2 + y 2 ≤ 3 4 . Find whether the integral of function f ( x , y ) is exist over the disk x 2 + y 2 ≤ 1 and provide reason for the result.

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Find the integral value of the function f(x,y)=1(1−x2−y2) over the disk x2+y2≤34. Find whether the integral of function f(x,y) is exist over the disk x2+y2≤1 and provide reason for the result.

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Chapter 14 Solutions

Find the integral value of the function f(x,y)=1(1−x2−y2) over the disk x2+y2≤34.

Find whether the integral of function f(x,y) is exist over the disk x2+y2≤1 and provide reason for the result.