Volumes of revolution

[/0]

Consider . If we find the volume when revolved around the x-axis from x = 0 to x = 3, the volume is


However, what if we did the problem another way... the average value of from x = 0 to x = 3 is 1.5.

If we say we get .

a) Why is this result different?
b) What value of '' would be necessary to achieve the same volume, and how would this be found without knowing the correct volume initially?

[dcc]

Because of the square. (i.e. The volume of a solid of revolution does not depend linearly upon the value of the function).

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Thanks dcc, I was suspicious about that.

Is there an 'average' that could be used though, like RMS or something else?

[zzdfa]

the problem is, 'average' is a very vague term:
http://en.wikipedia.org/wiki/Average
if you think about it, it usually means 'a value you can use the replace all the other values in a set and still get the same result in a certain calculation', which can sometimes simplify stuff. for example, if you were to eat 30 apples today and 90 apples tomorrow, you could use the arithmetic mean and replace the 30 and 90 by 60 and get the same result.

but this particular average doesn't work for say, rates of change. if you drive to work at 60km/h and drive back at 120km/h your average speed is not 90km/h.

so in this case:

b)'a cone with radius r  and height h has the same volume as a cylinder with the same height and what radius, ?'

so the answer is, the cylinder would have to have a radius of r/sqrt(3)

so in this sense you could define the 'average' radius of a cone as r/sqrt(3).
but this average isn't very useful/applicable to other problems, which is why you wont find it on wikipedia or anything.

[dcc]

I don't see any 'nice' way to get the same volumes in both.  I believe that functions for which this property holds naturally satisfy the following functional relationship:

.  

GOOD LUCK WITH THAT!