# Volume of generated solid by rotation

• ombudsmansect
In summary, the conversation discusses finding the volume of a solid generated when a curve is rotated about the x axis. The relevant formula for solving this is I = r^2 dm, which can be found in the Calculus bible. The final solution involves using (pi) integral of (f(x))^2 dx.
ombudsmansect

## Homework Statement

Find the volume of the solid generated where y = 1/x for 1<=x<=5 is rotated about the x axis.

## Homework Equations

I = r^2 dm, sqrt(1 + (dy/dx)^2) dx,

## The Attempt at a Solution

So I have found the length of the curve and can henceforth find surface area and so on but I cannot figure out the step to take to get to volume, I understand that it probably isn't too difficult after the first part, so does anyone knw what the relevant formula is to solve for this, any help much appreciated :) btw solid will b the shape of a spinning top in a way ty :)

ombudsmansect said:

## Homework Statement

Find the volume of the solid generated where y = 1/x for 1<=x<=5 is rotated about the x axis.

## Homework Equations

I = r^2 dm, sqrt(1 + (dy/dx)^2) dx,

## The Attempt at a Solution

So I have found the length of the curve and can henceforth find surface area and so on but I cannot figure out the step to take to get to volume, I understand that it probably isn't too difficult after the first part, so does anyone knw what the relevant formula is to solve for this, any help much appreciated :) btw solid will b the shape of a spinning top in a way ty :)

Hi again Ombudsmand,

The first thing that I notice here is

$$I = \int r^2 dm$$ which is the definition of the moment of ineteria..

Look this up in the Calculus bible and you will find what you are looking for..

Last edited:
hey :) Is ther really a calculus bible online?? lol but i can't find it in my textbook and all ones i find online are specific to certain shapes but I am sure that like the third integral or something like tht hsould do it. I can find the area under the curve then i know i have 2pi degrees of rotation but how to put that into volumetric terms within the given bounds? thanks again for helpin me out

ombudsmansect said:
hey :) Is ther really a calculus bible online?? lol but i can't find it in my textbook and all ones i find online are specific to certain shapes but I am sure that like the third integral or something like tht hsould do it. I can find the area under the curve then i know i have 2pi degrees of rotation but how to put that into volumetric terms within the given bounds? thanks again for helpin me out

Edwards and Penney...

Roting a solid about a fixed axis..

Should be a bell Ring :)

hmmm perhaps all i need is (pi) integral of (f(x))^2 dx, ill give it a shot and see what happens

lol! de ja vu frm reading that lol, well i used (pi) integral of (f(x))^2 dx and got the correct answer so all looks well. Thanks heaps again u r really good at helping ppl saved me twice tonite already! have a good one suz :D

ombudsmansect said:
lol! de ja vu frm reading that lol, well i used (pi) integral of (f(x))^2 dx and got the correct answer so all looks well. Thanks heaps again u r really good at helping ppl saved me twice tonite already! have a good one suz :D

You are welcome :)

## What is the definition of "volume of generated solid by rotation"?

The volume of a generated solid by rotation refers to the three-dimensional space that is enclosed by a shape that has been rotated around an axis. It is a measurement of the amount of space that the shape occupies.

## How is the volume of a generated solid by rotation calculated?

The volume of a generated solid by rotation can be calculated using the formula V = π∫a^b (f(x))^2 dx, where a and b are the limits of integration, f(x) is the function representing the shape, and π is the constant pi. This formula is derived from the disk method, where the shape is divided into infinitely thin disks and their volumes are summed together.

## What are the different methods for finding the volume of a generated solid by rotation?

There are two main methods for finding the volume of a generated solid by rotation: the disk method and the shell method. The disk method involves summing the volumes of infinitely thin disks, while the shell method involves summing the volumes of infinitely thin cylindrical shells. Both methods can be used depending on the shape of the solid and the axis of rotation.

## What types of shapes can be used to generate solids by rotation?

Any two-dimensional shape can be used to generate a solid by rotation, as long as it is rotated around a specific axis. Common shapes include circles, rectangles, triangles, and more complex curves and polygons. The shape must also have a defined boundary in order to be used in the calculation of the volume.

## What are some real-world applications of calculating the volume of generated solids by rotation?

The calculation of the volume of generated solids by rotation has several practical applications, such as in engineering, architecture, and manufacturing. For example, it can be used to determine the volume of a water tank or the capacity of a cylindrical container. It can also be used in designing and constructing objects with specific shapes and volumes, such as car engines or airplane wings.

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