Thread: Area under a curve (?)

1. 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 $n_0$ such that n $> n_0$, 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.
Thanks.

2. 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 $f$ is Riemann integrable, where $f(x)$ is (as you defined) the cross-sectional area of the solid at height $x.$

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