Tomorrow in calculus we’ll be discussing how to approximate volumes of solids of revolution using “shells”. This is traditionally a very difficult concept for students to visualize, so having a physical model is really helpful. The trouble is, no physical models seem to exist. We’ve all talked about “onion skins” or brought in Quarto pieces or other shell-like objects, or even cut cakes into shells for illustration, but those things are just simplified versions of the situation. Our volumes-by-shells model on Day 144 was nice, but made in Tinkercad and thus not very customizable. Today we upgrade to an OpenSCAD model. The model shown on the left is the solid obtained by revolving the region between the graph of y=4-x^2 and the x-axis on [0,2] around the y-axis, and the model on the right is an approximation of that solid using eight shells.
Now using the fact that a^2-b^2 = (a+b)(a–b), together with the definition of m_k and DeltaX = x_k – x_{k-1} and a bit of algebra, we can write this volume as:
This formula isn’t any easier to use in practice, say if we were to actually calculate and add up the volumes of the eight shells pictured above; however the form of this kth volume expression – and in particular the presence of the DeltaX in the expression – allows us to construct a definite integral that represents the exact volume of the solid of revolution.
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