Volume, moment, mass of solid of revolution

In summary, S is a solid of revolution 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. The questions involve finding the volume, moment My, and center of mass of the solid object S. To solve this, you will need to use the general triple integral formula and know how to find the surface area generated by a revolving curve. The surface area can be formulated as an integral, but it is not necessary to evaluate it at this time.
  • #1
kate45
12
0
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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  • #2
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?
 
  • #3
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
 

FAQ: Volume, moment, mass of solid of revolution

What is the volume of a solid of revolution?

The volume of a solid of revolution is the amount of space that is occupied by the solid when it is rotated around a given axis. It can be calculated using the formula V = π∫ab (f(x))2 dx, where f(x) is the function that defines the shape of the solid.

How do you find the moment of a solid of revolution?

The moment of a solid of revolution is the product of its volume and the distance from its center of mass to the axis of rotation. It can be calculated using the formula M = ∫ab (f(x))2 x dx, where f(x) is the function that defines the shape of the solid.

What is the mass of a solid of revolution?

The mass of a solid of revolution is the amount of matter that makes up the solid. It can be calculated by multiplying the density of the material by its volume. The density can be found by dividing the mass by the volume.

What is the difference between moment of inertia and moment of a solid of revolution?

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, while the moment of a solid of revolution is a measure of its volume and distance from the axis of rotation. Moment of inertia is typically used in physics and engineering, while the moment of a solid of revolution is used in calculus and geometry.

How can the mass of a solid of revolution be calculated accurately?

The mass of a solid of revolution can be accurately calculated by using techniques such as integration or numerical methods. The accuracy of the calculation depends on the precision of the measurements and the complexity of the shape of the solid. In some cases, it may be necessary to use 3D modeling software to obtain a more accurate mass calculation.

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