General Equations for Certain Volumes of Revolution

In summary, the volume of a cone can be calculated using the formula 1/3*B*H, where B is the base of the cone and H is its height. This can be understood by visualizing a cone as the rotation of the line y=x with respect to the y-axis. Similarly, the volume of a paraboloid can be calculated using the formula 1/2*B*H, where B is the base of the paraboloid and H is its height. This can be understood by visualizing a paraboloid as the rotation of the line y=x^2 with respect to the y-axis. By equating the two equations for volume and using the relationship H=ax^n, we can determine that the shape
  • #1
Pjennings
17
0
The volume of a cone =
1
- B H where B is the base of the cone and H is its height.
3

We can think about a cone as the line y = x rotated with respect to the y axis. The volume of a parabaloid =
1
- B H
2

and a parabaloid is the line y = x^2 rotated with respect to the y axis. So what if you wanted to know the what line has the equation 1/4*B*H or in general K*B*H, where K is a constant.
Let y= a*x^n and y=H, which is the height. Then x = (y/a)^1/n.

Then the integral for the volume of revolution becomes
[0]\int[/H](y/a)^1/n dy

Using a substitution and integrating we get that Volume= a*pi*((H/a)^((2+n)/n))/((2+n)/n)
We know that V = K*pi*x^2*H, and therefore we can equate the two equations for volume, and using the relationship that H= ax^n we get

a*pi*((H/a)^((2+n)/n))/((2+n)/n) = K*a*pi*x^2*x^n

Pi and a cancel out on both sides. Using the relationships x^2 =(y/a)^2/n and y=h we get


((H/a)^((2+n)/n))/((2+n)/n))= K*((H/a)^((2+n)/n))

and ((H/a)^((2+n)/n)) cancels out on both sides leaving

K= n/(n+2) where n is the value to which x is raised: x^n. Solving for n we get that n=2k/(1-k). Since we can think of a cone as the line y = x rotated with respect to the y-axis, then we need to let n=1 and see if k=1/3, which it does. You could also use this to determine what shape has the equation 1/4B*H by letting k=1/4, and you get n=2/3. So the line y= x^(2/3) rotated with respect to the y-axis has the equation 1/4*B*H

Having done all of this work, does anyone have any suggestions on where to go from here with this work?
 
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  • #2
Sorry, this is not the answer to your question, but the forum provides a really nice latex display (enclose it in "tex" tags) that would make your equations MUCH easier to read :)
 

FAQ: General Equations for Certain Volumes of Revolution

1. What is a volume of revolution?

A volume of revolution is a three-dimensional shape formed by rotating a two-dimensional shape around an axis. The resulting shape is also known as a solid of revolution.

2. What are some examples of volumes of revolution?

Some common examples of volumes of revolution include spheres, cylinders, cones, and tori (donuts). These shapes can be formed by rotating a circle, rectangle, triangle, or other two-dimensional shape around an axis.

3. How do I find the volume of a solid of revolution?

The volume of a solid of revolution can be calculated using specific equations depending on the shape being rotated. For example, the volume of a sphere can be found using the formula V = (4/3)πr^3, where r is the radius of the sphere.

4. How does changing the axis of rotation affect the volume of a solid of revolution?

The volume of a solid of revolution is directly affected by the axis of rotation. If the axis of rotation is changed, the resulting shape and therefore the volume will also change. For example, rotating a circle around its diameter will result in a cylinder, while rotating it around a tangent will result in a cone.

5. Can the volume of a solid of revolution be negative?

No, the volume of a solid of revolution cannot be negative. It is a measurement of space and therefore can only have positive values.

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