Explanation Given information: The centroid of the region in the first quadrant is bounded below by the x -axis, above by the curve y = ln x and on the right by the line x = e . Calculation: Express the integral as below. y = ln x Show the curve for the given function as in Figure (1). Calculate the moment of the region. M = ∫ 1 e y d x Substitute ln x for y . M = ∫ 1 e ln x d x Apply integral substitution by assuming the variable u as ( ln x ) . u = ( ln x ) d u d x = ln x d u = ln x d x Apply integral substitution by assuming the variable v as 1. d v = 1 Integrate the above equation. ∫ v = x Apply UV method. ∫ 1 e ln x d x = u v − ∫ v d u Substitute u for ( ln x ) , ln x d x for d u , x for v and 1 for ∫ v . ∫ 1 e ln x d x = [ ( x ) ln x − ∫ 1 d x ] = [ x ln x − x ] 1 e = [ ( e ln e − e ) − ( 1 ln 1 − 1 ) ] = [ ( e − e ) − ( − 1 ) ] = 1 Calculate the moment about x axis. M x = ∫ 1 e y 2 2 d x Substitute ln x for y . M x = ∫ 1 e ( ln x ) 2 2 d x Apply integral substitution by assuming the variable u as ( ln x ) 2 . u = ( ln x ) 2 d u d x = 2 ln x d u = 2 ln x d x Apply integral substitution by assuming the variable v as 1. d v = 1 Integrate the above equation. ∫ v = x Apply UV method. ∫ 1 e ( ln x ) 2 2 d x = ( ∫ 0 1 ( u v − ∫ v d u ) d x ) Substitute u for ( ln x ) 2 , 2 ln x d x for d u , x for v and 1 for ∫ v

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 8, Problem 21AAE

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Calculate the centroid of the region.

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Chapter 8, Problem 20AAE

Chapter 8, Problem 22AAE

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