Explanation The center of mass and centroid both are equivalent if the given body is of uniform density. The centroid is the point at which the mass of whole body centered i.e. if we suspend the body from that point, it will be perfectly balanced horizontally. If we have given two functions Formula used: A = ∫ a b f ( x ) − g ( x ) d x M x = ∫ a b 1 2 { f 2 ( x ) − g 2 ( x ) } d x M y = ∫ a b x { f ( x ) − g ( x ) } d x x _ = M y A , y _ = M x A Given: y = x + 1 and y = ( x − 1 ) 2 Calculation: First, we will sketch the graph and marked the common region. y = x + 1 is an equation of line passing through (0, 1) and (-1,0) and y = ( x − 1 ) 2 is an equation of parabola passing through (1,0) and symmetric about the line x=1 . So, let’s draw the graph The region enclosed by the given curves is the shaded region. To find the centroid, first we will calculate the area by the formula A = ∫ 0 3 f ( x ) − g ( x ) d x ∴ A = ∫ 0 3 { x + 1 − ( x − 1 ) 2 } d x ⇒ A = ∫ 0 3 ( x + 1 − x 2 − 1 + 2 x ) d x ⇒ A = ∫ 0 3 ( 3 x − x 2 ) d x ⇒ A = 3 x 2 2 − x 3 3 ] 0 3 ⇒ A = 3 ( 3 ) 2 2 − ( 3 ) 3 3 ⇒ A = 9 2 Now, the moments are M x = ∫ 0 3 1 2 { f 2 ( x ) − g 2 ( x ) } d x and M y = ∫ 0 3 x { f ( x ) − g ( x ) } d x ∴ M x = ∫ 0 3 1 2 { ( x + 1 ) 2 − ( x − 1 ) 4 } d x and M y = ∫ 0 3 x { ( x + 1 ) − ( x 2 − 2 x + 1 ) } d x ⇒ M x = 1 2 ...

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Author: Rogawski

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Chapter 9.4, Problem 30E

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Thecommon region and centroid of the region.

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Chapter 9.4, Problem 29E

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Let the region R be the area enclosed by the function f(x)= = 3x² and g(x) = 4x. If the region R is the base of a solid such that each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in the region R, find the volume of the solid. You may use a calculator and round to the nearest thousandth. y 11 10 9 00 8 7 9 5 4 3 2 1 -1 -1 x 1 2

Let the region R be the area enclosed by the function f(x) = ex — 1, the horizontal line y = -4 and the vertical lines x = 0 and x = 3. Find the volume of the solid generated when the region R is revolved about the line y = -4. You may use a calculator and round to the nearest thousandth. 20 15 10 5 y I I I | I + -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 -5 I -10 -15 I + I I T I I + -20 I + -25 I I I -30 I 3.5 4 x

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Chapter 9 Solutions

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Thecommon region and centroid of the region.

Solution Summary: The author explains that the center of mass and centroid are equivalent if the given body is of uniform density.

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