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: Region is between the graph y = x 2 and x-axis 0 ≤ x ≤ 2 Calculation: We have to find the area enclosed by the graph y = x 2 and x-axis, shown in the above figure. A = 1 2 ∫ O A ¯ x d y − y d x + 1 2 ∫ A B ¯ x d y − y d x + 1 2 ∫ C x d y − y d x -----------------------------(1) First we have to find line integrals along O A ¯ , A B ¯ and parabola C Let us now parameterize along O A ¯ x = t 0 ≤ t ≤ 2 and y = 0 Therefore, d x = d t and d y = 0 x d y − y d x = t ( 0 ) − 0 ( d t ) = 0 ∫ O A ¯ x d y − y d x = 0 --------------------------------(2) Let us now parameterize along A B ¯ x = 2 and y = t 0 ≤ t ≤ 4 Therefore, d x = 0 and d y = d t x d y − y d x = 2 ( d t ) − t ( 0 ) = 2 d t ∫ A B ¯ x d y − y d x = ∫ 0 4 2 d t = 2 [ t ] 0 4 = 2 ( 4 − 0 )

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 18.1, Problem 21E

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To calculate: area of the region between the graph $y = x^{2}$ and x-axis $0 \leq x \leq 2$

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Chapter 18.1, Problem 20E

Chapter 18.1, Problem 22E

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Chapter 18 Solutions

CALCULUS (CLOTH)

Ch. 18.1 - Prob. 1PQ Ch. 18.1 - Prob. 2PQ Ch. 18.1 - Prob. 3PQ Ch. 18.1 - Prob. 4PQ Ch. 18.1 - Prob. 5PQ Ch. 18.1 - Prob. 1E Ch. 18.1 - Prob. 2E Ch. 18.1 - Prob. 3E Ch. 18.1 - Prob. 4E Ch. 18.1 - Prob. 5E

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area of the region between the graph y = x 2 and x-axis 0 ≤ x ≤ 2

Solution Summary: The author explains how to calculate the area of the region between the graph y = x 2 and X-axis.

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To calculate: area of the region between the graph $y = x^{2}$ and x-axis $0 \leq x \leq 2$

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Chapter 18 Solutions

area of the region between the graph y = x 2 and x-axis 0 ≤ x ≤ 2

Solution Summary: The author explains how to calculate the area of the region between the graph y = x 2 and X-axis.

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To calculate: area of the region between the graph y=x2 and x-axis 0≤x≤2

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Chapter 18 Solutions

To calculate: area of the region between the graph $y = x^{2}$ and x-axis $0 \leq x \leq 2$