Minimal surfaces were in the display case

The Approximation of a Solid of Revolution

Most math teachers I've had have been able to break down Calculus into two very broad categories: derivatives and integrals. What is truly amazing, is how much you can do with these two tools. By using integration, it is possible to approximate the shape of a 2-D function that is rotated around an axis. This solid created from the rotation is known as a solid of revolution. To explain this concept, we will take a look at the region bounded by the two functions: \[ f(x) = 2^{.25x} - 1 \] and \[ g(x) = e^{.25x} - 1 \] bounded at the line y = 1. This region is meant to represent a cross section of a small bowl. While it may not perfectly represent this practical object, the approximation will be quite textured, and will provide insight into how the process works. The region bounded by the two functions can be rotated around the y-axis to create a fully solid object. This is easy enough to talk about, but what exactly does this new solid look like? Is

Area under a region

If you want to find the area under a region, you can draw rectangles and add the area of the rectangles together to find an aproximation. If you want to find the exact area, you use an integral bounded by the region. But what about when we introduce a 3rd plane? How do we aproximate the volume under a region when we consider all three planes? The concept is very similar. To aproximate the volume, we use rectangles but add the element of height and sum the rectangles volumes. For the actual volume, we use a double integral (using Fubini's Theorem) bounded by the region. Consider the region contained in the intersection of \[y=x^\frac{1}{2}\] \(and\) \[y=x^3.\] We are going to aproximate and find the actual volume of this region under the function \[f(x,y)=xy.\] We're going to find the aproximated area using squares, so let's zoom in a little bit until we have a 5x5 area for a total of 25 squares to work with. Do

Come visit the quadric surfaces!

They are in the display case. The ruled surfaces will be joining them soon!

MA 391 Composition and Communication

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