Explanation Given information: The density of the plate is δ . The region bounded by the graphs of y = x 4 and y = x 5 . Calculation: The equation of graph is y = x 4 (1) The equation of graph is y = x 5 (2) Calculate the intersection points as shown below. Substitute x 5 for y in Equation (1). x 5 = x 4 x 5 − x 4 = 0 x 4 ( x − 1 ) = 0 x = 0 or x = 1 Sketch the region bounded by the graphs of y = x 4 and y = x 5 as shown in Figure 1. Refer to Figure 1. Consider a vertical strip has center of mass ( x ˜ , y ˜ ) = ( x , x 4 + x 5 2 ) The length of strip is x 4 − x 5 and width of strip is dx . Calculate area of the strip as shown below. d A = length × width Substitute x 4 − x 5 for length and dx for width. d A = ( x 4 − x 5 ) d x Calculate the mass of the strip as shown below. d m = δ d A Substitute ( x 4 − x 5 ) d x for d A . d m = δ × ( x 4 − x 5 ) d x = δ ( x 4 − x 5 ) d x Calculate the mass of the plate as shown below. ∫ d m = ∫ 0 1 δ ( x 4 − x 5 ) d x M = δ ( x 5 5 − x 6 6 ) 0 1 = δ ( 1 5 5 − 1 6 6 − 0 ) = δ ( 6 − 5 30 ) = δ 30 Calculate the moment of the strip about x axis as shown below. y ˜ d m = ( x 4 + x 5 ) 2 δ ( x 4 − x 5 ) d x = δ 2 ( x 8 − x 10 ) d x Calculate the moment of the plate about x axis as shown below

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 6.6, Problem 21E

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Calculate the center of mass of a thin plate and show that it lies outside the region.

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Chapter 6.6, Problem 20E

Chapter 6.6, Problem 22E

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University Calculus: Early Transcendentals (4th Edition)

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