Volume, moment, mass of solid of revolution

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SUMMARY

The discussion focuses on calculating the volume, moment, and center of mass for a solid of revolution formed by rotating a specific curve around the y-axis. The solid is defined by the curve x = (2 - y)(y - 1)^2 within the bounds of y = 1 to y = 2. The volume is determined using the formula A(y) = π(2 - y)(y - 1)^2, which is integrated over the specified interval. Additionally, the formulation of the surface area integral is discussed, although the evaluation of this integral is not required.

PREREQUISITES
  • Understanding of solid of revolution concepts
  • Familiarity with triple integrals for calculating center of mass and moments
  • Knowledge of surface area calculations for revolving curves
  • Basic proficiency in calculus, specifically integration techniques
NEXT STEPS
  • Study the general triple integral formulae for evaluating center of mass and moments
  • Learn how to derive the surface area of solids of revolution
  • Practice integration techniques specific to volume calculations in 3D geometry
  • Explore applications of solid of revolution in engineering and physics
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Students and professionals in mathematics, engineering, and physics who are involved in geometric modeling and analysis of solids of revolution.

kate45
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Hi there,

I have no idea about this question can anyone help?

S is a solid of revolution in 3-dimensions, formed by rotating a full turn about the y-axis, the region in the first quadrant of the (x, y)-plane bounded by the interval [1, 2] on the y-axis, and the curve x = (2 − y)(y − 1)^2

(a) Find the volume, moment My and centre of mass of the solid ob ject S .
(b) Formulate as an integral, but do not evaluate, the surface area of solid S.
 
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You said you have no idea how to do the questions. Have you learned the general triple integral formulae for evaluating centre of mass, moments? And do you know how to find the surface area generated by a revolving curve?
 
HI,

I think i need to find A(y) which equals pi(2-y)(y-1)^2

then integrate this with respect to the intervals 1 and 2 which then gives the volume?

the moment and centre of mass, plus that last section i am stuck on
 

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