Explanation Given Information: The function, f ( x , y ) = x . The provided figure is, Formula used: The double integral over a nonrectangular, if R is the rectangle a ( y ) ≤ x ≤ b ( y ) and c ≤ y ≤ d , then ∫ ∫ f ( x , y ) d x d y = ∫ c d ∫ b ( y ) a ( y ) f ( x , y ) d x d y The equation of a line passing through the points ( x 1 , y 1 ) and ( x 2 , y 2 ) is y − y 1 y 2 − y 1 = x − x 1 x 2 − x 1 . Calculation: Consider the provided figure The figure represents the region in the first quadrant bounded by a line with intercepts ( 0 , 2 ) and ( 2 , 0 ) . Since, the two points are ( 0 , 2 ) and ( 2 , 0 ) . Substitute 2 for x 1 , 0 for y 1 , 0 for x 2 and 2 for y 2 in y − y 1 y 2 − y 1 = x − x 1 x 2 − x 1 . y − 0 2 − 0 = x − 2 0 − 2 Multiply both sides by 2. y − 0 2 − 0 × 2 = x − 2 0 − 2 × 2 y = − ( x − 2 ) = − x + 2 = 2 − x Thus, the equation is y = 2 − x . Thus, the upper limit and lower limit for y is 2 − x and 0 respectively and upper limit and lower limit for x is 2 and 0 respectively. The region R is 0 ≤ x ≤ 2 and 0 ≤ y ≤ 2 − x

Student Solutions Manual for Waner/Costenoble's Applied Calculus, 7th

7th Edition

ISBN: 9781337291293

Author: Waner, Stefan; Costenoble, Steven

Publisher: Brooks Cole

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Chapter 8.5, Problem 18E

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To calculate: The value of the integral ∬Rf(x,y)dxdy where f(x,y)=x and the figure describes the region R.

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Chapter 8.5, Problem 17E

Chapter 8.5, Problem 19E

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Chapter 8 Solutions

Student Solutions Manual for Waner/Costenoble's Applied Calculus, 7th

Ch. 8.1 - For each function in Exercises 14, evaluate (a)...Ch. 8.1 - For each function in Exercises 14, evaluate (a)...Ch. 8.1 - For each function in Exercises 14, evaluate (a)...Ch. 8.1 - For each function in Exercises 14, evaluate (a)...Ch. 8.1 - For each function in Exercises 5-8, evaluate (a)...Ch. 8.1 - Prob. 6E Ch. 8.1 - For each function in Exercises 5-8, evaluate (a)...Ch. 8.1 - For each function in Exercises 5-8, evaluate (a)...Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

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The value of the integral ∬ R f ( x , y ) d x d y where f ( x , y ) = x and the figure describes the region R .

Solution Summary: The author explains how to calculate the value of the integral displaystyleundersetRiint.

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To calculate: The value of the integral ∬Rf(x,y)dxdy where f(x,y)=x and the figure describes the region R.

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Chapter 8 Solutions