Approximating a Volume by Rectangular Prisms


Next, I modeled the volume under the curve \(z = 16 – x^2 – 2y^2\), using a double Riemann sum approximation. Here, in the \(xy\)-plane, each approximation contains \(m\) squares along the \(x\)-axis and \(n\) squares along the \(y\)-axisScreen Shot 2015-07-01 at 10.02.27 AMIn this case, I used \(m = n = 4\) and \(m = n = 8\).  In order to print the solids, I modeled them on Cinema 4D.  They were extremely thin (the height was much greater than the width and length) (see Cinema 4D pictures), so I divided the height of each approximation by 4.  I then printed them simultaneously, which took 8 hours and 45 minutes.  




The models turned out well, except in the \(m = n = 8\) case the printer did not print one of the rectangular prisms because I accidentally placed two rectangular prisms in the same place.   I then printed a c
orrected version as well as a model of the original solid (height divided by 4 once again).  On the original solid, I imprinted the equation onto the bottom, which does not look great because it should be imprinted deeper into the solid.  We considered using Magic Merge because the first print of the \(m = n = 8\) approximation printed some vertical lines, but when we fixed the model in Cinema 4D, the MakerBot printed smooth faces on each side.

The three objects can be found on Thingiverse  here, here, and here.