Explanation Given information: The region is bounded by the parabola y = 4 − x 2 and the line y = x + 2 . ρ = 1 Formula used: Let f and g be continuous and satisfy f ( x ) ≥ g ( x ) on the interval [ a , b ] , and consider a thin plate (lamina) of uniform density ρ that covers the region R between the graphs of y = f ( x ) and y = g ( x ) on the interval [ a , b ] . Then the mass of R is given by m = ρ ∫ a b ( f ( x ) − g ( x ) ) d x And the centroid of R is the point ( x ¯ , y ¯ ) such that x ¯ = M y m = ∫ a b x [ f ( x ) − g ( x ) ] d x ∫ a b [ f ( x ) − g ( x ) ] d x a n d y ¯ = M x m = 1 2 ∫ a b [ [ f ( x ) ] 2 − [ g ( x ) ] 2 ] d x ∫ a b [ f ( x ) − g ( x ) ] d x Calculation: The region R is bounded by the parabola y = 4 − x 2 and the line y = x + 2 . (Given) Let f ( x ) = 4 − x 2 and g ( x ) = x + 2 To find the point of intersection, f ( x ) = g ( x ) ⇒ 4 − x 2 = x + 2 and g ( x ) = 0 ⇒ x 2 + x − 2 = 0 ⇒ ( x + 2 ) ( x − 1 ) = 0 ⇒ x = − 2 , 1 Thus, the points of intersection are ( − 2 , 0 ) a n d ( 1 , 3 ) . The mass of R is given by m = ρ ∫ − 2 1 ( f ( x ) − g ( x ) ) d x = ∫ − 2 1 ( x 2 − x − 11 ) d x = ( x 3 3 − x 2 2 − 11 x ) − 2 1 = ( 1 3 − 1 2 − 11 ) − ( − 8 3 − 4 2 + 22 ) = ( − 6

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6.5, Problem 30PS

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To find:The centroid of the given planar region.

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Chapter 6.5, Problem 29PS

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Chapter 6 Solutions

Calculus

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The centroid of the given planar region.

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