Explanation Given information: The given function is y = 1 − x 2 and the interval is [ 0 , 1 ] . Density is ρ = 1 . 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 x -moment of a lamina S of density ρ occupying the region between the graphs of x = g 1 ( y ) and x = g 2 ( y ) over an interval [ a , b ] with g 1 ( y ) > g 2 ( y ) is given by M x = ρ ∫ a b y ( g 1 ( y ) − g 2 ( y ) ) d y The power rule of integral calculus is as shown. ∫ x n d x = x n + 1 n + 1 + C , n ≠ − 1 The following definite integral can be calculated as shown below. ∫ y 1 − y d y = 2 5 ( 1 − y ) 5 / 2 − 2 3 ( 1 − y ) 3 / 2 + C Calculation: Now let us calculate the x moment by the formula M x = 1 2 ρ ∫ a b [ f ( x ) ] 2 d x . Consider the given function. f ( x ) = 1 − x 2 Consider the given interval. [ 0 , 1 ] Here, a = 0 , b = 1 M x can be calculated as shown below

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Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 8.4, Problem 10E

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The x moment of the lamina of uniform density ρ occupying the region beneath the graph of y=1−x2.

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Chapter 8.4, Problem 9E

Chapter 8.4, Problem 11E

Chapter 8 Solutions

Calculus: Early Transcendentals

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The x moment of the lamina of uniform density ρ occupying the region beneath the graph of y = 1 − x 2 .

Solution Summary: The author explains the x moment of a lamina of uniform density, rho, over an interval, and the power rule of integral calculus.

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To find:

The x moment of the lamina of uniform density ρ occupying the region beneath the graph of y=1−x2.

Want to see the full answer?

Chapter 8 Solutions

The x moment of the lamina of uniform density ρ occupying the region beneath the graph of y = 1 − x 2 .

Solution Summary: The author explains the x moment of a lamina of uniform density, rho, over an interval, and the power rule of integral calculus.

Concept explainers

Videos

To find: The x moment of the lamina of uniform density ρ occupying the region beneath the graph of y=1−x2.

Want to see the full answer?

Chapter 8 Solutions

To find:

The x moment of the lamina of uniform density ρ occupying the region beneath the graph of y=1−x2.