Explanation Given information: The density of the plate is δ . The region bounded by the parabolas y = 2 x 2 − 4 x and y = 2 x − x 2 . Calculation: Provide the Equation of parabolas as shown below. y = 2 x 2 − 4 x (1) y = 2 x − x 2 (2) Sketch the region bounded by the parabolas y = 2 x 2 − 4 x and y = 2 x − x 2 as shown in Figure 1. Refer to Figure 1. The plate is symmetric about the line x = 1 . Hence, the center of mass x ¯ = 1 . Consider a vertical strip has center of mass ( x ˜ , y ˜ ) = ( x , ( 2 x − x 2 ) + ( 2 x 2 − 4 x ) 2 ) = ( x , x 2 − 2 x 2 ) Calculate the length of the strip as shown below. Length = ( 2 x − x 2 ) − ( 2 x 2 − 4 x ) = 2 x − x 2 − 2 x 2 + 4 x = 6 x − 3 x 2 = 3 ( 2 x − x 2 ) The width of strip is dx . Calculate area of the strip as shown below. d A = length × width Substitute 3 ( 2 x − x 2 ) for length and dx for width. d A = 3 ( 2 x − x 2 ) d x Calculate the mass of the strip as shown below. d m = δ d A Substitute 3 ( 2 x − x 2 ) d x for d A . d m = δ 3 ( 2 x − x 2 ) d x = 3 δ ( 2 x − x 2 ) d x Calculate the mass of the plate as shown below. ∫ d m = ∫ 0 2 3 δ ( 2 x − x 2 ) d x M = 3 δ ( 2 x 2 2 − x 3 3 ) 0 2 = 3 δ ( 2 2 − 2 3 3 − 0 ) = 3 δ ( 4 − 8 3 ) = 3 δ ( 4 3 ) = 4 δ Calculate the moment of the strip about x axis as shown below

Student Solutions Manual For Thomas' Calculus Format: Paperback

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ISBN: 9780134439334

Author: Hass, Joel R.^heil, Christopher D.^weir, Maurice D.^heil, Christopher

Publisher: PEARSON

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Chapter 6.6, Problem 16E

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