Explanation Given information: The region is bounded by the parabola y = x 2 − 9 and the x-axis. 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 = x 2 − 9 and the x-axis ( y = 0 ) . (Given) Let f ( x ) = x 2 − 9 and g ( x ) = 0 To find the point of intersection, f ( x ) = g ( x ) ⇒ x 2 − 9 = 0 and g ( x ) = 0 ⇒ x 2 = 9 ⇒ x = ± 3 Thus, the points of intersection are ( − 3 , 0 ) a n d ( 3 , 0 ) . The mass of R is given by m = ρ ∫ − 3 3 ( f ( x ) − g ( x ) ) d x = ∫ − 3 3 ( x 2 − 9 ) d x = ( x 3 3 − 9 x ) − 3 3 = ( 3 3 3 − 9 ( 3 ) ) − ( ( − 3 ) 3 3 − 9 ( − 3 ) ) = ( 9 − 27 )

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 6.5, Problem 27PS

To determine

To find:The centroid of the given planar region.

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Chapter 6.5, Problem 26PS

Chapter 6.5, Problem 28PS

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6. Find the maximum area of a rectangle in the first quadrant where one side of the rectangle lies along the line y = 10 – 2x and where the two vertices which are not on the given line are on the positive x- and y- axes.

Chapter 6 Solutions

Calculus

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The centroid of the given planar region.

Solution Summary: The author explains how to find the centroid of the given planar region. The region is bounded by the parabola and the x-axis.

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To find:The centroid of the given planar region.

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Chapter 6 Solutions