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Thread: Area under a curve (?)

  1. #1
    Junior Member
    Jan 2010

    Area under a curve (?)

    (First, I'm very sorry if this is in the wrong forum -- I'm taking a general "history of math" class, but I haven't taken any analysis/topology so I'm not sure if this question fits in.)

    Anyway, my question is: Take 3 parallel planes with a solid having bases on the top and bottom planes, each base being a simple closed curve. If the second plane moves and the area of the solid intersecting that plane is f(x), graphing f(x) and splitting into small segments of dx=h/n, find a sufficient condition for "for every epsilon, there exists some $\displaystyle n_0$ such that n$\displaystyle > n_0$, V1 - V2 < epsilon.

    I know that V1 is the sum of circumscribed rectangles under the curve and V2 is inscribed. Unfortunately I know little else. I'd really appreciate any help.
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  2. #2
    Junior Member nimon's Avatar
    Sep 2009
    Edinburgh, UK
    Hi kimberu,

    I think most of the helpers on this forum are averse to answering questions where the symbols used are not defined, so it would be a good idea to make your question as clear as possible. For example: what are V1,V2,h; which plane is the second plane; and splitting what into small segments?

    Having said that, I think I got the gist of what you were asking: when do the upper and lower estimates of the volume of the solid converge to the same value? Well that always happens when $\displaystyle f$ is Riemann integrable, where $\displaystyle f(x)$ is (as you defined) the cross-sectional area of the solid at height $\displaystyle x.$

    The most important things to read about are the conditions of Riemann integrability, which returns lots of hits in google. In particular, $\displaystyle f$ is Riemann integrable if $\displaystyle f$ is continuous over the height of the solid.

    And, out of interest, this is probably a calculus question, but that's beside the point really!

    Hope this helps, and enjoy the rest of your course.
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