Explanation Given: x 2 + y 2 + ( R z H ) 2 = R 2 , z ≥ 0 , R ≥ 0 , H>0 Formula used: V = ∭ W 1 d v z ¯ = 1 v ∭ W z d v x ¯ = 1 v ∭ W x d v y ¯ = 1 v ∭ W y d v Calculation: we find the volume of the half ellipsoid,first change into cylindrical coordinate. Since x 2 + y 2 + ( R z H ) 2 = R 2 r 2 + ( R z H ) 2 = R 2 ( R z H ) 2 = R 2 − r 2 R 2 z 2 = H 2 R 2 − H 2 r 2 z = H 2 R 2 − H 2 r 2 R V = ∭ W 1 d v = ∫ 0 2 π ∫ 0 R ∫ 0 H 2 R 2 − H 2 r 2 R r d z d r d θ = ∫ 0 2 π ∫ 0 R r [ z ] 0 H 2 R 2 − H 2 r 2 R d r d θ = ∫ 0 2 π ∫ 0 R r [ H 2 R 2 − H 2 r 2 R ] d r d θ = 1 R ∫ 0 2 π ∫ 0 R r [ ( H 2 R 2 − H 2 r 2 ) 1 2 ] d r d θ = 1 R ∫ 0 2 π [ − 1 3 H 2 ( H 2 R 2 − H 2 r 2 ) 3 2 ] 0 R d θ = 1 3 R H 2 ∫ 0 2 π ( H 2 R 2 ) 3 2 d θ = H 3 R 3 3 R H 2 [ θ ] 0 2 π = 2 π H R 2 3 z

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 16.5, Problem 22E

To determine

To find: The centroid of W where W is the upper half of the ellipsoid $x^{2} + y^{2} + {(\frac{R z}{H})}^{2} = R^{2}$ .where $z \geq 0, R \geq 0$ and H>0.

To show:The centroid is depend on H but not on R.

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Chapter 16.5, Problem 21E

Chapter 16.5, Problem 23E

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Calculus - Standalone book

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The centroid of W where W is the upper half of the ellipsoid x 2 + y 2 + ( R z H ) 2 = R 2 .where z ≥ 0 , R ≥ 0 and H>0. To show: The centroid is depend on H but not on R.

Solution Summary: The author explains how to find the centroid of W, where W is the upper half of the ellipsoid.

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To find: The centroid of W where W is the upper half of the ellipsoid $x^{2} + y^{2} + {(\frac{R z}{H})}^{2} = R^{2}$ .where $z \geq 0, R \geq 0$ and H>0.

To show:The centroid is depend on H but not on R.

Want to see the full answer?

Chapter 16 Solutions

The centroid of W where W is the upper half of the ellipsoid x 2 + y 2 + ( R z H ) 2 = R 2 .where z ≥ 0 , R ≥ 0 and H&gt;0. To show: The centroid is depend on H but not on R.

Solution Summary: The author explains how to find the centroid of W, where W is the upper half of the ellipsoid.

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To find: The centroid of W where W is the upper half of the ellipsoid x2+y2+(RzH)2=R2.where z≥0,R≥0 and H>0. To show:The centroid is depend on H but not on R.

Want to see the full answer?

Chapter 16 Solutions

The centroid of W where W is the upper half of the ellipsoid x 2 + y 2 + ( R z H ) 2 = R 2 .where z ≥ 0 , R ≥ 0 and H>0. To show: The centroid is depend on H but not on R.

To find: The centroid of W where W is the upper half of the ellipsoid $x^{2} + y^{2} + {(\frac{R z}{H})}^{2} = R^{2}$ .where $z \geq 0, R \geq 0$ and H>0.

To show:The centroid is depend on H but not on R.