Please help me understand the method of Archimedes to compute volume of a paraboloid.
How is paraboloid of $x^2$ curve has the half the volume of a cylinder circumscribed on it
Thanks
Actually i am reading this article which i am showing a fee lines here
"Since the disk is cylindrical, it should have the volume equal to (pi)R2h where R is the radius at that part of the paraboloid.
So for example, the cylinder that is at h height has radius of xh, at height 2h, the radius is x2 ... so on to height nh with radius xn. This matches with our general parabola equation given earlier, xn2 proportional to rh, when we consider the equation of the volume of the disks. Also the rth inscribed piece is proportional to (r - 1)h and the rth circumscribed piece is proportional to rh. Finally, the rth piece of the cylinder is proportional to nh [since all r pieces are uniform].
This means that we can set up the ratios of the size of the paraboloid pieces to the cylindrical pieces. In particular, for the inscribed cylinders:
If we sum up all the pieces we get:
What i don't understand are the equations above. I mean where did n^2 come from
Say $r=h=1$. Then the cylinder has volume $\pi r^2h=\pi$.
Now let's compute the volume of the paraboloid: $\int_0^1\pi x^2\operatorname dy=\pi\int_0^!y\operatorname dy=\pi [y^2/2]_0 ^1=\pi/2$.