# Volumes by Slices: Iterated Integrals

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.