Explanation Given: ( x a ) 2 + ( y b ) 2 = 1 Let, the x -coordinate be ' 0 ' . For the y -coordinate, we integrate between the two points where it crosses the x -coordinate. Then we divide by the area. Now, the area of the ellipse is the product of the value of π , major axis and minor axis. When, y = 0 , ( x a ) 2 + ( 0 b ) 2 = 1 ( x a ) 2 = 1 x = a When, x = 0 , ( 0 a ) 2 + ( y b ) 2 = 1 y = b Therefore, major and minor axis of the ellipse is b and a respectively. Area of the ellipse, = π × ( b × a ) = π a b Area of the half ellipse, = 1 2 π a b Now, ( x a ) 2 + ( y b ) 2 = 1 ( y b ) 2 = 1 − ( x a ) 2 y 2 = b 2 ( 1 − x a 2 2 ) On integration, we ge ∫ y 2 d x = ∫ − a a b 2

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is…

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Chapter 9.3, Problem 29E

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To findthe centroid of the top half of the ellipse, ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} = 1$ .

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Chapter 9.3, Problem 28E

Chapter 9.3, Problem 30E

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Chapter 9 Solutions

Calculus - Standalone book

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To findthe centroid of the top half of the ellipse, ( x a ) 2 + ( y b ) 2 = 1 .

Solution Summary: The author explains that the area of the ellipse is the product of , major axis and minor axes.

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To findthe centroid of the top half of the ellipse, ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} = 1$ .

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Chapter 9 Solutions

To findthe centroid of the top half of the ellipse, ( x a ) 2 + ( y b ) 2 = 1 .

Solution Summary: The author explains that the area of the ellipse is the product of , major axis and minor axes.

Concept explainers

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To findthe centroid of the top half of the ellipse, (xa)2+(yb)2=1.

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Chapter 9 Solutions

To findthe centroid of the top half of the ellipse, ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} = 1$ .