Gabriel’s Horn
I know I said in my first Math Post that I would cover Euler’s Identity next, but I just came across this beauty, so I had to share it. This one’s called Gabriel’s Horn - a solid cone that has infinite surface area, but finite volume. An interesting little (or infinitely big) occurrence straight from the world of improper integrals. Let’s go ahead and dive into the math of it all:
- First we start by letting R be the region bounded by the graphs y = 1/x and y = 0 for x > 0
- then we get the area by computing the integral:
- this can be interpreted as the area being infinite or increasing without bound. So we’ve got infinite area, let’s see what happens when we rotate this function along the x-axis and get a 3D solid
it looks a litle like this:
(as a side note it looks like I can't just paste an equation I made from a google doc in here, so I wound up having to export the google doc as a pdf, take a screen clipping to make it an image and upload it here. It didn't come out as pretty as I'd like, but at least the equations are a little more presentable than in the first math post.) Once again, thank you all for reading.
-Carlos
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