Calculating Volume from Revolved Area

In summary, calculating volume from revolved area involves finding the area of a cross section and multiplying it by the distance the cross section is rotated around an axis. This method can be applied to various shapes, including circles, rectangles, and triangles, by using their respective formulas for area. The resulting volume measurement is useful in many real-world applications, such as determining the volume of a container or the amount of material needed for construction projects. It is a fundamental concept in geometry and is essential for understanding three-dimensional shapes and their properties.
  • #1
eberg
10
0
I remember calculating volume by revolving shapes around axis using integration in school. I have an arbitrary shape shape that has no easily attainable equation, if I know the area of that shape, can I calculate the volume?

This is an example. I know this area, and want the volume if revolved around the y-axis:

http://img149.imageshack.us/img149/1/shapepo1.th.png

-Eric
 
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  • #2
eberg said:
I have an arbitrary shape shape that has no easily attainable equation, if I know the area of that shape, can I calculate the volume?

Hi Eric! :smile:

No … compare the volume of a sphere with the volume of a cylinder with the same cross-section area.

Mathematically, volume = π∫(f(z))2dz, and area = ∫f(z)dz, and the integral of (f(z))2 isn't a fixed multiple of the integral of f(z). :smile:

hmm … how about the surface area of the solid? :wink:
 
  • #3
OK, thanks for the response.
 
  • #4
There is a theorem (I can't quite remember the name right now) that the volume of a solid of rotation is equal to the area of region rotated and the circumference of the circle swept out by the centroid of the region. Of course, without knowing the formula for the boundary of the region you can't find that circumference exactly but for reasonably shaped regions you might be able to approximate it by seeing where lines from opposite points intersect.
 
  • #5
HallsofIvy said:
There is a theorem (I can't quite remember the name right now) that the volume of a solid of rotation is equal to the area of region rotated and the circumference of the circle swept out by the centroid of the region. Of course, without knowing the formula for the boundary of the region you can't find that circumference exactly but for reasonably shaped regions you might be able to approximate it by seeing where lines from opposite points intersect.

Hi HallsofIvy! :smile:

(i think you omitted the word "product" :wink:)

yes, the centroid would be at a distance of ((1/2)∫(f(z))2dz)/area,

so the volume is 2π times that times the area. :smile:
 
  • #6
Thanks, tiny-tim.
And I now recall that it is "Pappus' Theorem".
 
  • #7
good ol' Pappus of Alexandria …

HallsofIvy said:
Thanks, tiny-tim.
And I now recall that it is "Pappus' Theorem".

Hi HallsofIvy! :smile:

At first, I thought, what's he talking about :rolleyes: … isn't Pappus' theorem that projective hexagon thing?

Then I looked in http://en.wikipedia.org/wiki/Pappus'_Theorem#Theorems", and found that good ol' Pappus :smile: had two theorems …
Although Pappus's Theorem usually refers to Pappus's hexagon theorem, it may also refer to Pappus's centroid theorem.

so you're absolutely right! :biggrin:
 
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  • #8
Remarkable!
 
  • #9

FAQ: Calculating Volume from Revolved Area

What is the formula for calculating volume from revolved area?

The formula for calculating volume from revolved area is V = π ∫ab [f(x)]2 dx, where a and b are the limits of integration and f(x) is the function representing the area being revolved.

What is the difference between volume and revolved area?

Volume is a measure of the amount of space occupied by a three-dimensional object, while revolved area is the surface area of a two-dimensional shape that is rotated around an axis.

Can volume be calculated from any revolved area?

Yes, volume can be calculated from any revolved area as long as the shape being revolved is symmetrical and the limits of integration are known.

How can I use calculus to calculate volume from revolved area?

Calculus can be used to calculate volume from revolved area by applying the disk method or the shell method. The disk method involves slicing the revolved shape into infinitely thin disks and integrating their volumes, while the shell method involves slicing the revolved shape into infinitely thin shells and integrating their volumes.

Are there any real-world applications for calculating volume from revolved area?

Yes, calculating volume from revolved area has many real-world applications, such as in engineering, architecture, and manufacturing industries. It is also commonly used in physics and mathematics to solve problems involving rotational symmetry.

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