If this can't be found, which famous mathematician(s) worked on further developing the idea of finding volumes through revolution?
Who invented finding volumes through revolution
calculusmath-historysolid of revolutionvolume
Related Solutions
Just to offer a closure to this question, this may benefit from the inclusion of diagrams. Erlend appears to be proposing the use of "disks", but it should be keep in mind that disk "slices" are always perpendicular to the rotation axis. Since that is "vertical" in this problem, the slices are "horizontal" and so will have "thicknesses" $ \ dy \ . $ So we will need to integrate in the $ \ y-$ direction, which requires expressing the disk radii as a function of $ \ y \ . $
The function inversion for the parabola is $ \ x = \sqrt{\frac{y}{a}} \ $ , and this is applicable over the entire interval in the $ \ y-$ direction, $ \ 0 \le y \le h \ , $ so the volume integral is
$$ \int_0^h \ \pi \ [x(y)]^2 \ \ dy \ = \ \pi \ \int_0^h \ \frac{y}{a} \ \ dy \ = \ \frac{\pi}{ a} \ \left( \ \frac{y^2}{2} \vert_0^h \ \right) \ = \ \frac{\pi \ h^2}{2 \ a} \ , $$
as already shown by Jean-Claude Arbaut.
[The apparent discrepancy in "dimensionality" is due to the fact that $ \ a \ $ has the "dimension" of inverse length. Our result is correctly a three-dimensional volume.]
Some of Erlend's confusion may be arising from the requirements for applying the "shell method." For that, the shell-wall "slices" are always parallel to the rotation axis.
The shell "thicknesses" are now $ \ dx \ , $ so the integration is carried out in the $ \ x-$ direction, which will use the single interval $ \ 0 \le x \le \sqrt{\frac{h}{a}} \ , $ as he was writing (but for the wrong method). It is the integrand which must be "split", as the "height" of the shells extends from the parabola "upward" to the "horizontal" line $ \ y = h \ . $ Here, we express the function for the parabola in terms of $ \ x \ , $ making the shell heights $ \ h - ax^2 \ . $
The radii of the shells again extend perpendicularly from the rotation axis, so these are given by $ \ r = x \ . $ The shell volume integration is thus
$$ \int \ 2 \pi \ r \ h \ \ dr \ \ \rightarrow \ \ 2 \pi \ \int_0^{\sqrt{\frac{h}{a}}} \ x \ \cdot \ (h - ax^2) \ \ dx $$
$$ = \ 2 \pi \ \int_0^{\sqrt{\frac{h}{a}}} \ hx - ax^3 \ \ dx \ = \ 2\pi \ \left( \ \frac{hx^2}{2} \ - \ \frac{ay^4}{4} \ \vert_0^{\sqrt{h/a}} \ \right) $$
$$ = \ 2 \pi \ \left( \ \frac{h}{2} \cdot \frac{h}{a} \ - \ \frac{a}{4} \cdot \frac{h^2}{a^2} \ \right) \ = \ 2 \pi \ \cdot \ \frac{h^2}{4a} \ = \ \frac{\pi \ h^2}{2 \ a} \ , $$
as found above.
GeoGebra and WolframAlpha (thanks @user170231!) seem to work well for me right now. Here's a quick overview of how solids of revolution can be graphed using either software:
WolframAlpha
Explain the figure in as precise a language as possible. For example, "The solid formed by rotating the area bound by the functions $x^2$ and $\sqrt{x}$ about the x axis". Type this into the text field at WolframAlpha. You should get a graph!
There is one downside to this though. The graph can't be moved or rotated, which may not allow for a good intuition all the time. Since this is powered by AI, it's not guaranteed to work all the time either. GeoGebra offers a more comprehensive graphing platform. GeoGebra's downside, however, is that it's impossible (or really difficult) to graph rotations like this. Instead, only the rotations of curves can be sketched, which means that a problem such as the one above must be split into two. Instead of graphing the rotation of the area between the functions $x^2$ and $\sqrt{x}$, we must graph the rotations of both the functions separately and mentally imagine subtracting the volume of one from the other.
GeoGebra
Go to the GeoGebra 3D Calculator. I will demonstrate the graphing functionality with the same example that was used above. Note that $0\leq x\leq 1$ should be replaced with the intersections of both the functions chosen, and the functions themselves ($x^2$ and $\sqrt{x}$) must be replaced.
- f(x)=If(0<=x<=1,x^2)
- g(x)=If(0<=x<=1,sqrt(x))
- a=Surface(f,360deg)
- b=Surface(g,360deg)
$a$ and $b$ can be switched on or off and colors changed to give a better intuition of the solid.
And of course, the graph can be moved around and rotated to give a better sense of what the solid really looks like! I found these tools really helpful for visualizing solids of revolution, thus helping me solve problems involving these better. They also helped me understand the intuition behind the formulas for finding the volume of such solids. I hope this was helpful for you too!
Best Answer
I think you mean Pappus of Alexandria. See