Explanation Given information: The given function is y = x 3 and the interval is [ 0 , 2 ] . Formula used: The x -moment of a lamina S of density ρ occupying the region beneath the graph of y = f ( x ) over an interval [ a , b ] , is given by M x = 1 2 ρ ∫ a b [ f ( x ) ] 2 d x The y -moment of a lamina S of density ρ occupying the region beneath the graph of y = f ( x ) over an interval [ c , d ] is given by M y = ρ ∫ c d x f ( x ) d x The area of a lamina S of density ρ occupying the region beneath the graph of y = f ( x ) over an interval [ a , b ] , is given by A = ∫ a b f ( x ) d x Centroid of a lamina S lying underneath the graph of a function y = f ( x ) over an interval [ a , b ] , is given by ( x ¯ , y ¯ ) = ( M y M , M x M ) Where M = ρ A The power rule of integration is as shown below. ∫ x n d x = x n + 1 n + 1 + C , n ≠ − 1 Calculation: The lamina of a constant density ρ occupying the region beneath the graph of y = x 3 for 0 ≤ x ≤ 2 can be constructed as shown below. Consider the given function. f ( x ) = x 3 Consider the given interval. [ 0 , 2 ] Here, a = 0 , b = 2 M x can be calculated as shown below. M x = 1 2 ρ ∫ a b [ f ( x ) ] 2 d x = 1 2 ρ ∫ 0 2 [ x 3 ] 2 d x = 1 2 ρ ∫ 0 2 x 6 d x = 1 2 ρ [ x 7 7 ] 0 2 = 64 7 ρ M y can be calculated as shown below

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ISBN: 9781319545956

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 8.4, Problem 9E

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The x, y moments and the center of mass of the lamina of uniform density ρ occupying the region beneath the graph of y=x3.

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Chapter 8.4, Problem 8E

Chapter 8.4, Problem 10E

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The x , y moments and the center of mass of the lamina of uniform density ρ occupying the region beneath the graph of y = x 3 .

Solution Summary: The author explains how to find the x, y moments and the center of mass of a lamina of uniform density.

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To find:

The x, y moments and the center of mass of the lamina of uniform density ρ occupying the region beneath the graph of y=x3.

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Chapter 8 Solutions

The x , y moments and the center of mass of the lamina of uniform density ρ occupying the region beneath the graph of y = x 3 .

Solution Summary: The author explains how to find the x, y moments and the center of mass of a lamina of uniform density.

Videos

To find: The x, y moments and the center of mass of the lamina of uniform density ρ occupying the region beneath the graph of y=x3.

Want to see the full answer?

Chapter 8 Solutions

To find:

The x, y moments and the center of mass of the lamina of uniform density ρ occupying the region beneath the graph of y=x3.