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I've browsed around the forum a bit and found that others have had the same problem as me, however, none of the answers help me a lot, so I thought to post a more specific question, I hope you don't mind.

I'm having a problem with the surface area of a sphere, probably because I'm extending "backwards", ie. if the derivative of sphere volume is sphere surface, why is the integral over the circumference of a circle not the surface area of a sphere? More formally, why is this:

[tex]2\int_0^R 2\pi r dr \neq 4\pi R^2[/tex]

I know plenty of other ways of obtaining the surface area of a sphere, also through integration, but I simply lack the geometric understanding to show me why this particular integral does NOT describe the surface area of a (hemi)sphere.

Can anyone enlighten me? Bear in mind, I am by no means very good at math. I know a bunch of tools, and how to use them, but my understanding of them is vague at best, and that is what I am trying to rectify.

Thank you in advance. :-)