Volume of a bead with square hole- Program approach?

[DM48]

Description

1 - We have sphere of radius R with rectangular prism removed from it. Centerline of prism includes center of sphere.



2 - Length of one side (along x axis) of rectangle is a length of another side (along y axis) is b.

3 - The following schematic represents cross-sections of sphere with the void in 3 planes.



The problem is to find the volume of sphere with the rectangular prism removed through center.

Solution

1 - Volume is found by subtracting volume of prism with spherical top and bottom from volume of sphere.

\( V=V_{sphere}-V_{prism}=\frac{4}{3}\,\pi \, r^{3}-V_{prism} \)

2 - The prism volume can be found by integration. For this purpose, let's divide the prism in 8 parts of equal volume the same way the simple prism is divided.




3 - The volume of part of prism with spherical ends can be found by integrating equation which describes upper part of sphere’s surface (the one which has positive z coordinates). The equation in Descartes coordinates is z(x,y)=√(r^2-x^2-y^2 ), assuming that sphere center is point 0,0,0. So we can write:

Descartes coordinates \( z(x,y)=\sqrt{r^2-x^2-y^2} \)

\( V_{prism}=8\int_{0}^{\frac{b}{2}}\int_{0}^{\frac{a}{2}}\sqrt{r^2-x^2-y^2}\,dx\,dy \)


\( V_{prism}=4\int_{0}^{\frac{b}{2}}(r^2-y^2)\,arcsin\left (\frac{a}{2\sqrt{r^2-y^2}} \right )\,dy\,+2a\int_{0}^{\frac{b}{2}}\sqrt{r^2-\frac{a^2}{4}-y^2}\,\,dy \)


\( V_{prism} = 4\int_{0}^{\frac{b}{2}}\left ( r^2-y^2 \right )\,arcsin\left ( \frac{a}{2\sqrt{r^2-y^2}} \right )\,dy\,+\,\frac{a}{4}\left ( 4r^2-a^2 \right )arcsin\left ( \frac{b}{\sqrt{4r^2-a^2}} \right )\,+\,\frac{ab}{4}\sqrt{4r^2-a^2-b^2} \)


4 - The integral in the right side of equation above can be represented as:

\( \int_{0}^{\frac{b}{2}}\left ( r^2-y^2 \right )\,arcsin\,\left ( \frac{a}{2\sqrt{r^2-y^2}} \right )\,dy \)

Next part will take time and I'm not sure I can do it.


Also, unsure if this is correct but it gets us going in a direction. I wanted to push this out to get opinions. I will rework through this several more times and see what I come up with.

Best way to proceed on a program like this with Free42/DM42? Hoping for Werner and Albert to chime in.