I've scribbled in purple the region of integration needed to find the volume of the steinmetz 3 solid. I also know the bounds needed to find the volume of the steinmetz solid. I just can't picture how the region I'm shading is 1/4 the volume of the entire steinmetz tricylinder solid. Can someone clarify how the region I'm drawing is actually the an abstraction of the steinmetz 3 solid?
If you integrated using this above region, you'd have to multiple your final integral by 16 to get the answer
I can't imagine how the region I'm shading is abstraction of the actual steinmetz solid given above credits to wolfram http://mathworld.wolfram.com/SteinmetzSolid.html
I know we could also integrate using this region and multiply the final integral by 16 but this still doesn't help me picture why this is the volume of the steinmetz solid (tryclinder equal radii intersection)
Let's do something simpler: look at the bottom picture you included. There's a vertex $P$ nearest to us, right? Let's suppose that all three of its coordinates are positive (indeed, I guess they're all $1/sqrt{3}$ or something like that...maybe $1/\sqrt{2}$?)
And $P$ is "joined" by "edges" to three other vertices. Let's call those $A,B,C$. Draw in edges $AB$, $BC$, and $CA$, And join each of those arcs to the origin, forming a triangle with an arc-edge.
Then everything within the solid and ALSO in the octant whose bounding quarter planes include those three triangles, (i.e., the part of the solid it's easiest for us to see in your picture!) constitutes one eighth of the solid you care about. (Because it's the intersection of the solid with the octant $x, y, z \ge 0$, and there are exactly 8 octants defined by various signs for $x,y,z$.)
The thing you've scribbled on (in the middle picture) consists of two of these octants, hence is 1/4 of the total volume. (The top picture seems to be showing a larger scribbled-on region.)
Here's a picture showing the points $P, A, B, C$ and the three arcs (in yellow).