Explanation Formula Used: Area enclosed by C = ∮ C x d y = ∮ C y d x = 1 2 ∫ C x d y − y d x Given: Vertices of square are (1,1) (-1,1) (-1,-1) and (1,-1) Calculation: We have to find the area enclosed by square ABCD shown in the above figure. A = 1 2 ∫ A B ¯ x d y − y d x + 1 2 ∫ B C ¯ x d y − y d x + 1 2 ∫ C D ¯ x d y − y d x + 1 2 ∫ D A ¯ x d y − y d x -----------------------(1) First we have to find line integrals along A B ¯ , B C ¯ , C D ¯ and D A ¯ Let us now parameterize along A B ¯ x = t , 1 ≥ t ≥ − 1 and y = 1 d x = d t and d y = 0 x d y − y d x = t ( 0 ) − 1 ( d t ) = − d t Therefore, ∫ A B ¯ x d y − y d x = ∫ 1 − 1 − d t ∫ A B ¯ x d y − y d x = − [ t ] 1 − 1 = − [ − 1 − 1 ] ∫ A B ¯ x d y − y d x = 2 --------------------------------(2) Let us now parameterize along B C ¯ x = − 1 and y = t , 1 ≥ t ≥ − 1 d x = 0 and d y = d t x d y − y d x = − 1 ( d t ) − t ( 0 ) = − d t ∫ B C ¯ x d y − y d x = ∫ 1 − 1 − d t ∫ B C ¯ x d y − y d x = − [ t ] 1 − 1 = − [ − 1 − 1 ] ∫ B C ¯ x d y − y d x = 2 ---------------------------------(3) Let us now parameterize along C D ¯ x = t , − 1 ≤ ...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

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Chapter 18.1, Problem 22E

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To calculate: area of the square.

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Chapter 18.1, Problem 21E

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Chapter 18 Solutions

CALCULUS (CLOTH)

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Solution Summary: The author explains how to calculate the area enclosed by square ABCD.

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Chapter 18 Solutions