lecture18

lecture18 - MATH 100 Lecture 18 Triple Integral, II;...

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2006 Fall MATH 100 Lecture 8 1 MATH 100 Lecture 18 Triple Integral, II; centroid { } {} 2 1 2 1 12 (,) (,,) (,) , or , Then the orders are (,,) and gx z z GR gy z z R x y z g xz yg xz R R x y z zx gy z y zR f x y z dV f x y z dy dA f x y z dV f x y z dx dA =≤ ⎡⎤ = ⎢⎥ ⎣⎦ = ∫∫∫ ∫∫ ∫ Class Triple integral, centroids Integration in other orders
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2006 Fall MATH 100 Lecture 8 2 MATH 100 Lecture 18 Triple Integral, II; centroid
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2006 Fall MATH 100 Lecture 8 3 MATH 100 Lecture 18 Triple Integral, II; centroid V M = δ Center of gravity of a solid: Solid G homogeneous (uniform composition & structure) Solid G inhomogeneous material V M z y x v Δ Δ = Δ 0 lim ) , , (
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2006 Fall MATH 100 Lecture 8 4 MATH 100 Lecture 18 Triple Integral, II; centroid G ** * 11 Mass of G ( , , ) lim lim ( , , ) nn kn n n k kk xyzdV M xyz V δ →∞ == = = Δ ∫∫∫ ∑∑ Center of gravity: 1 (,,) Mass of G 1 Mass of G 1 Mass of G G G G xx x yzdV yy x yzdV z zx = = =
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2006 Fall MATH 100 Lecture 8 5 For homogeneous, ( , , ) (a const), we have centroid of the region 1 Vol of G 1 Vol of G 1 Vol of G G G G xyz xx d V yy d V zz d V δ = = = ∫∫∫
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2006 Fall MATH 100 Lecture 8 6 MATH 100 Lecture 18 Triple Integral, II; centroid Explanation of center of gravity: Consider a point-mass in located at x , then the tendency for the mass to produce a rotation about a point a on the axis is measured by the following quantity: moment of m
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This note was uploaded on 09/17/2011 for the course MATH 100 taught by Professor Qt during the Fall '09 term at HKUST.

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lecture18 - MATH 100 Lecture 18 Triple Integral, II;...

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