I have modeled the the solid from Example 8 of Section 12.1 from Stewart’s Essential Calculus. It is bounded by the surfaces \(z=\sin x \cos y\), \(z=0\), \(y=0\), and \(x=\pi/2\). The example demonstrates the strategy behind computing a double integral using Fubini’s Theorem. I approximated the solid by eight slices in the \(x\) and \(y\) directions. In order to draw the correct splines in Cinema 4D, I had to use the correct parametric equations to plug into the inputs \(x(t),\, y(t)\), and \(z(t)\). For the first object I held \(x\) constant (approximating integration with respect to the \(y\)-variable). The parametric equations were \(x(t)=\frac{k\pi}{32} \), \(y(t)=t\), and \(z(t)=\sin(\frac{k\pi}{32}) \cos(t)\) for \( k=1, 3, 5, \dots, 15 \). I then created a slice in Cinema 4D by adding straight splines. I extruded each slice by 0.2, which is just greater than \(\pi/16\) (the width of each slice). I placed each slice so that it overlapped slightly with the next slice – this will allow the objects to be merged (via a Boole) and will prevent vertical lines from showing in the print after the slices are collected into one object. I had to use a Boole with two cubes for the two approximations by slices because the solids each had two very thin edges (that is, I shaved off some volume from two of the edges).

I then repeated the entire process, but this time with a constant \(y\)-variable. I also printed the smooth solid, which is the volume that is approximated by the slices. In order to do this, I imported the solid from Mathematica and put equations on it like I have in other models. These models can be found on Thingiverse here, here, and here.