Do Over: Integral over Regions in the Plane

Have you ever completed a project, turn it in, and then immediately think of all the ways you could have improved upon it and you wish you could just do it over? Well, I was able to do just that. When creating a model that demonstrates how integrating over the region of a plane works, there are many routes you can take. I decided to make an approximation of what was essentially the volume of half a cone. I thought it was sufficient in showcasing how approximating the value of an integral works, but I knew I was limited by my tools. I had to manually create each rectangular prism that made up the approximation, which was kinda boring to be honest. It was so boring, that I only did the minimum number neccessary. I knew it would be more interesting if the approximation was made up of hundreds of prisms. That way, it would be more accurate to the real integral, but you could still see that it was an approximation. I wanted to do something like that, but I was not about to spend hours making rectangles for my weekly homework.
  The idea of making hundreds of prisms became a reality when I was introduced to OpenSCAD. With OpenSCAD, I could write ten lines of code, click a button, and boom. Hundreds of rectangular prisms right before my eyes. And that's exactly what I did for my do over project. Adding more rectangles wasn't the only improvement I made for this project. My original cone only grew in size as you move up the \(y\) axis. I wanted to have an integration over an area whose value doesn't just increase as you move in one direction. So, I introduced sine waves to the function. This way, the value of the function goes up and down as you go in any direction. The function being integrated for this project is \(f(x,y) = 3+\sin(x)+\sin(y)\).

Now, which object looks cooler? A blob of rectangles that you can't even tell is supposed to be a cone, or the area full of hills and valleys made from over a thousand rectangular prisms? Personally, I'd say the latter. I made the redo multichromatic so that you can see the hills easier. Ultimately, the increase in rectangular prisms and the change of function really improve this project in displaying the value of an integral over an area.