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1. I was attempting to find the surface area of a hemisphere by summing up the surface areas of infinitesimal cylinders (or ribbons) of increasing radii (excluding the top and bottom areas). I wanted to solve this integration problem this way as opposed to using spherical coordinates (or any of the other numerous methods).

2. The attempt at a Solution.

My method for solving this problem is exactly the same IBY describes in the following post: https://www.physicsforums.com/showthread.php?t=427695

Dick explains why this attempted solution is incorrect. By claiming: "It gives you the area of the ribbon if it's vertical (i.e. parallel to the direction dx). If not it ignores the horizontal component. That's not a good enough approximation."

The value that my attempted solution outputs shows that the solution I tried is incorrect; however, this brought up a question for which I have no understanding. What makes an integration a "good enough approximation"? I had thought that since the ribbons' heights are infinitesimal there would be no problem, but there is. Why, and how would one predict that there would be a problem?

If dick's explanation is correct, then why does integration ever converge to the area under a curve? For example if one was to find the area of under y=x (a 45 degree line) for any discrete change in x, the rectangles whose sum results in the area under the curve is not quite accurate. Since the change in x ceases to be discrete in integration (it becomes infinitely small), that problem is solved. Why is it not so with the above problem?

Thank you very much! I appreciate any help I can get.

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# When is integration good enough

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