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Jungle Speed vs Solids with Known Cross Sections

As a kid, I played a game called Jungle Speed. Here's a picture for reference, but essentially everyone playing flipped one card over at a time and when a specific pattern appeared, your goal was to grab the yellow stick before anyone else could. Well, when I thought about calculating the volume of a solid with known cross sections, the first thing that came to mind was that yellow stick. I know that sounds really random, but that's what I decided to try and recreate.

The cross sections of the yellow stick are circles, but in order to better explain cross sections I chose to use rectangular prisms as my cross sections as they are easy to visualize and have a basic volume equation. I now had to think about what equations could make up the wave like sides of the yellow stick and I figured a trig function would be best suited for a wave like shape. I chose the following functions and evaluated the volume of known cross sections of squares between \(x=0\) and \(x=10\):

\[f(x) = cos(x) + 2\] and \[g(x) = sin(x)\]

In order to approximate the volume of my solid with square cross sections, I had to calculate the volume of each of my ten rectangular prisms and add them together. Since the volume of a rectangular prism with a square base is \(V=b^2*h\), I calculated the difference between \(f(x)-g(x)\) to be my base and used a height of \(1\) for each prism to ensure an even spread. I used the midpoint of each cross section to get the most acurrate volume of my object. Below is a table showing a summary of my calculations and the total approximate volume of \(39.71units^3\):



In order to calculate the actual volume of my solid, I had to calculate the integral of the area of each cross section with respect to \(x\). The integral below shows the actual volume of my solid is about \(40.17units^3\):

\begin{aligned}\int _{0}^{10}\left[ \left( \cos x+2\right) -\sin x\right] ^{2}dx\\ .\end{aligned} \begin{aligned}=\int _{0}^{10}\cos ^{2}\left( x\right) +400s\left( x\right) -2sin\left( x\right) \cos \left( x\right) -4\sin \left( x\right) +sin^{2}\left( x\right) +4dx\\ =-\sin ^{2}\left( 10\right) +4\sin \left( 10\right) +4\cos \left( 10\right) +46=40.17units^{3}\end{aligned}

The solid that I ended up creating looked a little less like the yellow stick from my game and a little more like when the kid you babysit doesn't understand physics and tries to make a tower with their blocks. But, you can definitely still see the wave like sine and cosine functions that I wanted to see, so it wasn't a total fail. The image below shows the solid I will attempt to print. Once I created my solid, I scaled it down to 40% to create the dimensions \(101.7mm\) x \(40.28mm\) x \(34.69mm\).
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