Question 587934: Find the volume of a one-base paraboloid having on its base a circumference of 16π cm and an altitude of 10 cm.
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website! I'm not sure my solution is what your calculus teacher expects, but hopefully it will help.
I like to visualize that paraboloid in x-y-z coordinates (in cm) as the space between
 and  (I'll find the right  later).
The vertex of that paraboloid is at (0,0,0), with  .
At we have the base of the paraboloid, which is a circle.
Its radius, R is such that  --> 
If we were allowed, I would search for volume, V, of a paraboloid with height h and base radius a, and find the formula
 and for your paraboloid 
In some calculus class, I would have some formula for volume of revolution solids, but I d not have it. (My 1973 Tom Apostol Calculus book has accumulated too much dust, and is probably too heavy and abstract to be of use).
So I figure out the radius, r, of a cross section as a function of z.
At ,  is  --> 
Now I have  --> 
Then, the area of a cross section circle is  .
Now I'm ready to find the volume as
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