I Confusion on the Volumes of Solids of Revolution

[grafs50]

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.

 

[haruspex]

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.

 

[HallsofIvy]

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"?).

 

[haruspex]

Forget my previous response. I realize I may not have understood this part:

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?