Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20090831 03:48:36
Integration, what is it?Hi; Last edited by bobbym (20090831 03:55:20) In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #2 20100329 05:29:20
Re: Integration, what is it?I had a look and I liked it. Whilst the "area under the curve" is good, the idea of "multiplication for things that vary" could help some who can't apply concepts away from the way they are taught. I also suspect that it does provide the "ah" moment. #3 20100329 08:29:40
Re: Integration, what is it?Thinking of integration as an area is certainly a useful technique when it works, but the problem is that there are some integrable functions where that doesn't make sense. Why did the vector cross the road? It wanted to be normal. #4 20100329 09:52:29
Re: Integration, what is it?Hi random_fruit; What is the value of that integral? In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #5 20100329 10:25:32
Re: Integration, what is it?The integral of that function on any interval is 0. Why did the vector cross the road? It wanted to be normal. #6 20100329 10:59:32
Re: Integration, what is it?Hi mathsyperson; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #7 20100329 12:58:40
Re: Integration, what is it?
That integral does not exist as a Riemann integral. We can take the limit as Δx goes to zero by choosing all rationals for Δx, in which case each term in the sequence is 1, or we can take the limit by choosing Δx to be irrationals, in which case each term in the sequence is 0. This prove the limit does not exist. Where a_i is a value the simple function takes, E_i = f^{1}(a_i), and m(E_i) is the measure of E_i. This uses a bit of a fancy definition, but in this case you're helped a bit by the theorem that the measure of any countable set is 0, and m(A U B) = m(A) + m(B) when A and B are disjoint. This means the measure of the rationals in [0,1] is 0, and so your function has integral 0. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #8 20100406 04:36:10
Re: Integration, what is it?An example where area is not directly relevant is the work done in charging a capacitor. See http://en.wikipedia.org/wiki/Capacitance for more information. 