Confusion on the Volumes of Solids of Revolution

In summary, the conversation discusses the difficulty in using the average value of a function to determine the volume of a solid of revolution. The speaker first tries to find the volume of a solid of revolution around y=√x from 0 to 1 around the x-axis, but then attempts to solve for the volume of a solid of revolution around y=2/3 using a cylinder with radius 2/3 and height. However, this approach does not work and they are unsure why. The conversation also mentions the theorem of Pappus and the concept of average distance from the axis.
  • #1
grafs50
16
0
I've been trying to figure out why you can't use the average value of a function to determine the volume of a solid of revolution.

As an example:

Trying to find the volume of a solid of revolution on y=√x from 0 to 1 around the x-axis.
The definite integral is 2/3, which divided by one is still 2/3 so it is the average value.

Then I tried to use turn this into a problem of finding the volume of a solid of revolution around the y=2/3. a radius of 2/3 from 0 to 1. I Since the sides were a straight line I tried to solve for the volume of a cylinder with radius 2/3 and height.

But this didn't work. I've been wracking my brain for like half an hour trying to figure out why but I've got nothing. Can anyone explain to me why this doesn't work?

Thanks in advance.
 
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  • #2
It depends what you mean by "average radius". The average height of your y=√x is 2/3, but if you consider the elements in the volume of revolution, that is not their average distance from x axis. There are relatively more elements further from the axis.
 
  • #3
Perhaps you are thinking of the (second) "theorem of Pappus" (also called the "theorem of Guidinus") that the volume of a solid of revolution is equal to the area of the region revolved times the distance from traveled by the centroid of the region ("average"?).
 
  • #4
Forget my previous response. I realize I may not have understood this part:
grafs50 said:
Then I tried to use turn this into a problem of finding the volume of a solid of revolution around the y=2/3. a radius of 2/3 from 0 to 1. I Since the sides were a straight line I tried to solve for the volume of a cylinder with radius 2/3 and height.
If it's a rotation about y=2/3, what range of x is being rotated?
 

1. What is the concept of "volumes of solids of revolution"?

The concept of "volumes of solids of revolution" involves rotating a two-dimensional shape around an axis to create a three-dimensional shape, and then finding the volume of that shape.

2. How do you calculate the volume of a solid of revolution?

The volume of a solid of revolution can be calculated using the formula V = π∫(f(x))^2 dx, where f(x) is the function that defines the shape and the integral is taken over the desired interval of x values.

3. What are some common shapes used in finding volumes of solids of revolution?

Some common shapes used in finding volumes of solids of revolution include cylinders, cones, spheres, and more complex shapes such as toroids and spheroids.

4. How does the method of cylindrical shells differ from the method of disks and washers in finding volumes of solids of revolution?

The method of cylindrical shells involves slicing the solid into infinitely thin cylindrical shells and finding the volume of each shell, then adding them together. The method of disks and washers involves slicing the solid into infinitely thin disks or washers and finding the volume of each one, then adding them together. The method of cylindrical shells is generally used for more complex shapes, while the method of disks and washers is used for simpler shapes.

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

No, the volume of a solid of revolution cannot be negative. It represents the amount of space occupied by the solid, so it must always be a positive value.

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