Explanation Refer Fig. P5.21 and Fig. 1. Let the element of volume of the disk be of radius r and thickness dx . Find the equation for radius from the equation of the generating curve. x = h − h a 2 y 2 r 2 = a 2 h ( h − x ) Here, the y component is equal to the radius, h is the radius of the paraboloid, x is the centroid of the element and d x is the thickness of the element. Write the expression for the volume of the element. d V = π r 2 d x = π a 2 h ( h − x ) d x (I) Find the centroid of the element. x E L ¯ = x Here, x E L ¯ is the centroid of the element. Write the expression for the x ¯ for the volume. x ¯ V = ∫ x E L ¯ d V x ¯ = ∫ x E L ¯ d V V (II) Conclusion: Refer Fig.2 Calculate the volume of component 1. V 1 = ∫ 0 h / 2 d V = ∫ 0 h / 2 π a 2 h ( h − x ) d x = π ( a 2 h ) [ h x − x 2 2 ] 0 h / 2 = 3 8 π a 2 h Calculate ∫ 1 x E L ¯ d V for component 1. ∫ 1 x E L ¯ d V = ∫ 0 h / 2 x [ π a 2 h ( h − x ) d x ] = π ( a 2 h ) [ h x 2 2 − x 3 3 ] 0 h / 2 = 1 12 π a 2 h 2 Substitute 1 12 π a 2 h 2 for ∫ 1 x E L ¯ d V and 3 8 π a 2 h for V 1 in equation (II). x 1 ¯ = 1 12 π a 2 h 2 3 8 π a 2 h = 2 9 h Thus, the value of x ¯ for the first component is 2 9 h

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Chapter 5.4, Problem 5.124P

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To determine the value of x¯ for the two volumes.

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